Design and Discovery of New Piezoelectric Materials Using Density Functional Theory

DESIGN AND DISCOVERY OF NEW PIEZOELECTRIC MATERIALS USING DENSITY FUNCTIONAL THEORY

by Sukriti Manna © Copyright by Sukriti Manna, 2018 All Rights Reserved A thesis submitted to the Faculty and the Board of Trustees of the Colorado School of Mines in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy (Mechanical Engineering).

Golden, Colorado Date

Signed: Sukriti Manna

Signed: Dr. Cristian V. Ciobanu Thesis Advisor

Signed: Dr. Vladan Stevanovi´c Thesis Advisor

Golden, Colorado Date

Signed: Dr. John Berger Professor and Head Department of Mechanical Engineering

ii ABSTRACT

Piezoelectric materials find applications in microelectromechanical systems (MEMS), such as surface acoustic wave (SAW) resonators, radio frequency (RF) filters, resonators, and energy harvesters. Using density functional theory calculations, the present study illus- trates the influence of alloying and co-alloying with different nitrides on piezoelectric and mechanical properties of an existing piezoelectric material such as aluminum nitride (AlN). Besides improving the performance of existing piezoelectric material, a high-throughput screening method is used to discover new piezoelectric materials. AlN has several beneficial properties such as high temperature stability, low dielectric permittivity, high hardness, large stiffness constant, high sound velocity, and complementary metal-oxide-semiconductor (CMOS) compatibility. This makes it widely accepted material in RF and resonant devices. However, it remains a challenge to enhance the piezoelectric modulus of AlN. The first part of this thesis establishes that the piezoelectric modulus of AlN could be improved by alloying with rocksalt transition metal nitrides such as scandium nitride (ScN), yttrium nitride (YN), and chromium nitride (CrN). As the content of the rocksalt end member in the alloy increases, the accompanying structural frustration enables a greater piezoelectric response. This structural frustration is also accompanied by thermodynamic driving forces for phase separation which, with increased alloy concentration, lead to the destruction of the piezoelectric response upon transition to the (centrosymmetric, cubic) rocksalt structure. Thus, it becomes necessary to identify suitable alloying elements that may yield highest piezoelectric response with minimal alloying additions. The study reveals that the alloying with CrN would lead to the lowest transition composition (occurring at approximately 25% CrN concentration) between the wurtzite and rocksalt structures, thereby allowing piezoelectric enhancements at alloying levels that are easier to stabilize during the synthesis. The present study indicates that, for 25% CrN alloying, the piezoelectric modulus

iii is about 4 times larger than that of pure AlN. Thus, it is proposed to use CrxAl1−xN as a suitable replacement for AlN. One of the adverse eﬀect of transition metal nitride alloying for improvement of piezoelectric response is accompanied by the softening of AlN lattice. This would make the material less desirable for resonant applications. Our subsequent research establishes that co-alloying with YN and BN would enable the most superior combination of piezoelectric and mechanical properties of AlN. The rest of the thesis describes our eﬀorts for identifying new piezoelectric materials. This requires high-throughput screening of inorganic materials for their piezoelectric properties. Our quest to search for new piezoelectric compounds is motivated by the notion that soft materials have the opportunity to exhibit large piezoelectric response (piezoelectric modulus d), and the knowledge that the van der Waals (vdW) layered materials are typically soft in the layer stacking direction. From a pool of 869 vdW structures, we have discovered

50 new compounds such as In2Te5, GeTe, and CuVO3 having piezoelectric response larger than AlN. Remarkably, 70% of our discovered piezoelectric candidates do not contain any toxic elements like Pb. Our analysis reveals that the large components of d always couple with the deformations (shearing or axial) of van der Waals “gaps” between the layers. This research establishes a wide scope of synthesizing vdW materials for applications demanding high piezoelectric modulus.

iv TABLE OF CONTENTS

ABSTRACT ...... iii

LISTOFFIGURES ...... ix

LISTOFTABLES ...... xiii

LISTOFSYMBOLS...... xiv

LISTOFABBREVIATIONS ...... xv

ACKNOWLEDGMENTS ...... xvii

DEDICATION ...... xix

CHAPTER1 INTRODUCTION ...... 1

1.1 PiezoelectricEﬀectandApplications ...... 2

1.2 Classiﬁcation of Piezoelectric Materials ...... 3

1.3 Vector/Tensor Quantities Related to Piezoelectricity ...... 4

1.4 ConstitutiveEquations ...... 5

1.5 FiguresofMeritofPiezoelectricDevices ...... 7

1.5.1 FiguresofMerit...... 7

1.5.2 PiezoelectricDevices ...... 9

1.6 Piezoelectric Materials: Present Status and Motivation for Computational Investigations ...... 14

1.6.1 MaterialsbasedonSingleCrystal ...... 15

1.6.2 Materials Based on Polycrystalline Materials ...... 18

1.6.3 Toxicity of Lead-based Piezoelectrics ...... 23

v 1.6.4 Motivation for Computational Research on Improving Piezoelectric Materials ...... 23

1.6.5 Prior Computational Studies of Piezoelectric Properties...... 25

1.7 Objectives...... 26

1.7.1 Improving Performance of Existing Piezoelectric Materials ...... 27

1.7.2 Searching for New Piezoelectric Materials...... 27

1.8 OrganizationoftheThesis ...... 28

CHAPTER 2 ENHANCED PIEZOELECTRIC RESPONSE OF AlN via CrN ALLOYING ...... 31

2.1 Introduction...... 32

2.2 Methods...... 34

2.2.1 Paramagnetic Representation of Cr-AlN Alloys...... 34

2.2.2 DetailsoftheDFTCalculations ...... 34

2.2.3 ExperimentalProcedures...... 36

2.3 ResultsandDiscussion ...... 36

2.3.1 EnthalpyofMixing...... 36

2.3.2 PiezoelectricStressCoeﬃcients ...... 37

2.3.3 Comparison with Other Wurtzite-Based Alloys ...... 42

2.3.4 ExperimentalResults...... 46

2.4 ConcludingRemarks ...... 48

CHAPTER 3 TUNING THE PIEZOELECTRIC AND MECHANICAL PROPERTIES OF THE AlN SYSTEM VIA ALLOYING WITH YN ANDBN ...... 52

3.1 Introduction...... 52

3.2 Methods...... 55

vi 3.3 ResultsandDiscussions ...... 57

3.4 Conclusion...... 64

CHAPTER 4 LARGE PIEZOELECTRIC RESPONSE OF VAN DER WAALS LAYEREDSOLIDS ...... 66

4.1 Introduction...... 66

4.2 ComputationalMethodology...... 70

4.2.1 Automated Identiﬁcation of Quasi-2D Materials ...... 70

4.2.2 Calculating Piezoelectric Properties of Quasi-2D Materials ...... 71

4.2.3 Workﬂow for identifying quasi-2D piezoelectrics ...... 74

4.3 ResultsandDiscussions ...... 75

4.3.1 Promising Quasi-2D Piezoelectric Materials ...... 75

4.3.2 RoleofvanderWaalsInteractions ...... 80

4.3.3 Axial Piezoelectric and Elastic Anisotropy ...... 82

4.4 Conclusions ...... 84

CHAPTER 5 SUMMARY & CONCLUSIONS ...... 86

5.1 Conclusions ...... 86

5.2 RecommendationsforFutureWork ...... 87

REFERENCESCITED ...... 88

APPENDIX A CLASSIFICATION OF PIEZOELECTRIC MATERIALS ...... 110

APPENDIX B DENSITY FUNCTIONAL THEORY METHODS ...... 112

B.1 Many-bodySchr¨odingerequation ...... 112

B.2 DensityFunctionalTheory ...... 113

B.2.1 Hohenberg-KohnTheorem ...... 113

vii B.2.2 TheKohn-ShamEquations ...... 114

B.2.3 LocalDensityApproximation ...... 115

B.2.4 GeneralizedGradientApproximation ...... 116

B.3 Technicalimplementation ...... 116

APPENDIXC SIMULATINGELASTICPROPERTIES ...... 117

APPENDIX D MODELING OF RANDOM ALLOYS ...... 118

D.1 DetailsforGeneratingSQSStructures ...... 118

D.2 Dispersion of e33 and C33 values for diﬀerent Y concentration ...... 119

APPENDIX E ADDITIONAL SUPPLEMENTARY INFORMATION OF CHAPTER3 ...... 121

E.1 Variations of e31 and C11 withYandBcomposition ...... 121

E.2 Piezoelectric moduli as functions of composition ...... 121

APPENDIX F SUPPLEMENTARY INFORMATION OF CHAPTER 4 ...... 124

F.1 Comparison of Experimental and Calculated PiezoelectricModulus . . . . . 124

F.2 Axial Anisotropies: C vs e ...... 125

F.3 Experimentally Measured Elastic Constants of Quasi-2D Materials...... 125

F.4 Schematics of Diﬀerent Modes of d ...... 126

F.5 Rotation of d, e, and C -matrices ...... 127

APPENDIX G PERMISSIONS FOR CONTENT REUSED ...... 129

viii LIST OF FIGURES

Figure 1.1 Schematics of direct piezoelectric effects: (a) at applied compressive stress, (b) at applied tension. (c) and (d) describe the schematics of converse piezoelectric effect at applied electric field...... 3

Figure 1.2 Classiﬁcation of piezoelectric materials. They are classiﬁed into of piezoelectric, pyroelectric, and ferroelectric materials...... 4

Figure1.3 CorrelationbetweendiﬀerentFOMs ...... 13

Figure 1.4 The advancement and milestones of piezoelectric singlecrystals ...... 15

Figure 1.5 α-Quartz used as a piezoelectric oscillator in a watch ...... 17

Figure 2.1 Schematics of cation sublattice of CrxAl1−xN alloy. Al (Cr) sites are shown as gray (green) spheres. At a given Cr concentration, the Cr sites of each conﬁguration have a diﬀerent and random spin initialization with zero total spin in order to capture the paramagnetic state...... 35

Figure 2.2 (a) DFT-calculated mixing enthalpies of the wurtzite and rocksalt phases of CrxAl1−xN as functions of Cr concentration. (b) Calculated mixing enthalpies for several wurzite-based nitride alloys grown experimentally...... 38

Figure 2.3 The 18 components of the piezoelectric tensor calculated for SQS supercells with different spin configurations and compositions of Cr. At each Cr composition, the open circles represent data for each SQS cell used, while the red solid circles represent the average values across the randominitialconfigurations...... 39

total clamped nonclamped Figure 2.4 Variation of (a) e33 , (b) e33 component of e33, (c) e33 , (d) 33 component of the Born eﬀective charge tensor, (e) strain sensitivity of internal parameter, (f) c/a ratio with respect to Cr addition...... 40

Figure 2.5 (a) Variation of internal parameter, u with Cr addition. (b) Crystal structureofwurziteAlN...... 41

Figure 2.6 Comparison of change in e33 with addition of diﬀerent transition metals for x 25%regime...... 42 ≤

Figure 2.7 Variation of (a) C33 and (b) d33 for several nitride-based wurtzite alloys. . 43

ix Figure 2.8 Variation of e33 as function of Cr concentration for wurtzite phase alloysupto50%...... 45

Figure 2.9 Representative transmission electron microscopy (TEM) image (a) and an energy dispersive spectroscopy (EDS) line scan (b) of an (Al1−xCrx)N ﬁlm cross section containing 7% Cr, conﬁrming the incorporation of Cr into wurtzite solid solution.∼ EDS data were collected along the dashed black line shown in panel (a)...... 47

Figure 2.10 X-ray diffraction patterns of the thin film combinatorial libraries plotted against the film composition, with comparison to the patterns for wurtzite (WZ) and rocksalt(RS) structures. For alloying content x< 30%, the films grow predominantly with the wurtzite structure. At higher Cr concentrations, x> 30%, both rocksalt and wurtzite phases are detected, and the wurtzite exhibits degraded texture. No films were produced with compositions in regions where no intensity is shown. . . . 48

Figure 3.1 Atomic structure of a representative SQS Al50Y25B25N alloy structure . . 54

Figure 3.2 [(a) and (b)] Lattice constants a and c for AlN-YN and AlN-BN alloys as a function of dopant (Y or B) cation concentration x. Panel (c) shows the c/a ratioforthesealloys...... 55

Figure 3.3 Calculated mixing enthalpy curves for AlN alloyed with YN in the wurtzite(w)androcksalt(rs)structures...... 57

Figure 3.4 The design premise behind co-alloying AlN with YN and BN. (a) Higher YN content in Al1−xYxN increases the piezoelectric coefficient e33 but decreases the elastic modulus C33. (b) Increasing concentration of BN in Al1−yByN has a small effect on the piezoelectric coefficient, but increases the elastic modulus. Thus, co-alloying with YN and BN could lead to increases in both piezoelectric coefficient and stiffness relativetopureAlN...... 58

Figure 3.5 Variation of piezoelectric coeﬃcient in Al1−xYxN with strained applied along the a and c axes...... 59

Figure 3.6 The key piezoelectric (e33, red, left vertical axis) and elastic (C33, green, right axis) properties of Al-Y-B nitrides as functions of the Y concentration at constant boron levels ranging from (a) 0.00% B to (h) 43.75%B...... 61

2 2 Figure 3.7 The coupling constants (a) k33 and (b) k31 as functions of the Y and B concentrations in Al1−x−yYxByNwithatleast50%Al...... 63

x Figure 4.1 Illustration of a quasi-2D material (TaSe2), showing the atomic structure of the layers and the van der Waals spacing between them. . . . 68

Figure 4.2 Correlation between experimentally measured longitudinal piezoelectric modulus (d33) and their elastic modulus (C33) ...... 69

Figure 4.3 Elastic constants of quasi-2D materials calculated with (a) GGA and (b) vdW-corrected functional. The calculated values are compared with experimental ones. The calculations with the vdW-corrected functional are all within 50% error relative to the measurements. Details of measurement techniques, measurement temperatures, etc., are provided inthesupplementaryinformations...... 72

Figure 4.4 Workflow for calculations of piezoelectric coefficient tensor eij, the elastic stiffness Cij, and the piezoelectric modulus tensor dij of quasi-2D materials. A vdW-corrected functional (optB86) is used in all calculations. We have calculated piezoelectric modulus of 80 quasi-2D materials...... 73

Figure 4.5 Left panel: plot of d = max( d ) against the space group number max | ij| for 63 quasi-2D vdW materials with dmax > 0 (out of 135 non-centrosymmetric compounds). The two horizontal dotted lines denote the d33 values of PbTiO3 (119 pC/N) and AlN (5.5 pC/N). The blue square in this plot represents the hypothetical structure of PbS (distorted NiAs found in ICSD), not the ground rocksalt structure. Right panel: crystal structures of quasi-2D materials with large piezoelectricmoduli...... 75

Figure 4.6 Histogram of the maximum components of (a) the piezoelectric coeﬃcient tensor, emax and (b) piezoelectric modulus tensor, dmax. The most frequent emax is e33, whereas for dmax the most frequent maximum values are d15, d24 and d33. (c) Schematics of several important piezoelectric operating modes with their corresponding deformation types. The schematics of other relevant piezoelectric operating modes, i.e., components of dij are shown in the supplementary information. . . . 79

Figure 4.7 Correlation between axial anisotropy, i.e., out-of-plane to in-plane response ratio) of (a) d and e (b) d and C. The results reveal that majority of quasi-2D piezoelectric materials are dominated by out-of-plane piezoelectric responses (both in e and d) but elastically theyaredominatedinin-planedirection...... 83

Figure A.1 Centrosymmetric and non-centrosymmetric (polar and non-polar) point groups. Only non-centrosymmetric point groups allow piezoelectricity. . 110

xi Figure D.1 Calculated e33 and C33, for each individual SQS and averaged, at each concentration...... 119

Figure E.1 Piezoelectric coeﬃcient e31 and elastic modulus C11 as functions of Y andBconcentrations...... 122

Figure E.2 Calculated d33 and d31 surface in yttrium-boron composition space . . . 123

Figure F.1 Calculated d33 with GGA functional vs experimentally measured d33 . . 124

Figure F.2 Axial anisotropy in C vs axial anisotropy in e ...... 125

Figure F.3 Schematics of the modes of d with their measuring polarization directions and deformation stresses. Each mode represents each component of d-matrix ...... 126

Figure F.4 (a) ‘old’ and ‘new’ co-ordinate frames. (b) The angular relations between the axes is speciﬁed by drawing up a table of direction cosines which is denoted by a-matrix...... 127

xii LIST OF TABLES

Table 1.1 Piezoelectric devices and relevant FOMs. Each type of piezoelectric device has its own FOMs which characterize their performances...... 11

Table 1.2 Merits and demerits of several commercially used piezoelectric materials . . 24

Table 2.1 Piezoelectric properties of AlN and a few wurtzite alloys for piezoelectric deviceapplications...... 46

Table 3.1 Comparison between the properties of some of the Al-Y-B nitrides computed in this work and those of three commercial piezoelectrics. . . . . 62

Table 4.1 List of top 20 candidate quasi-2D piezoelectric materials are shown with their space group (SG) number, calculated DFT band gap (Eg), maximal piezoelectric modulus dmax, the dij component that appears as dmax, maximal eij (emax), the emax component of eij...... 77

Table F.1 Reference table for Figure 6. The measured values of Cij are reported in GPaunit...... 125

xiii LIST OF SYMBOLS

Electricﬁeldintensityorﬁeldstrength ...... E

Polarization or electric moment per unit volume ...... P

Electricdisplacement...... D

Straintensor ...... ǫ

Stresstensor ...... σ

Dielectricsusceptibility...... χ

Dielectricpermittivityofvacuum...... ǫ0

Stiﬀnesstensor ...... C

Compliancetensor ...... S

Identitymatrix ...... I

Piezoelectric strain modulus or piezoelectric modulus tensor ...... d

Piezoelectric stress coeﬃcient or piezoelectric coeﬃcienttensor...... e

Piezoelectricvoltageconstant ...... g

Energytransmissioncoeﬃcient ...... λmax

Curietemperature ...... TC

Mechanicalqualityfactor ...... Q or QM

Electromechanicalcouplingconstant ...... k

Acousticimpedance ...... Z

Dielectricconstant ...... ε

xiv LIST OF ABBREVIATIONS

AlternatingCurrent ...... AC

RadioFrequency ...... RF

SurfaceAcousticWave...... SAW

BulkAcousticWave ...... BAW

IntegratedCircuit ...... IC

PiezoelectricTransformer ...... PT

GlobalPositioningSystem ...... GPS

MicroelectromechanicalSystem ...... MEMS

FigureofMerit ...... FOM

WirelessLocalAreaNetworks...... WLAN

CompoundAnnualGrowthRate ...... CAGR

Complementary Metal-Oxide-Semiconductor ...... CMOS

Lead Zirconate Titanate or Pb(Zr,Ti)O3 ...... PZT

PbMg1/3Nb2/3O3 ...... PMN

Lithium niobate or LiNbO3 ...... LN

Lithium tantalate or LiTaO3 ...... LT

La3Ga5SiO14 ...... LGS

VisualInformationFidelity ...... VIF

DensityFunctionalTheory...... DFT

K0.5Na0.5NbO3 ...... KNN

xv Bi4Ti3O12 ...... BIT

Pb(Zn1/3Nb2/3)O3 ...... PZN

MorphotropicPhaseBoundary ...... MPB

TransmissionElectronMicroscopy ...... TEM

Wurtzite...... WZ

Rocksalt...... RS

Vienna ab initio SoftwarePackage ...... VASP

xvi ACKNOWLEDGMENTS

Pursuing a Ph.D. is difficult and sometimes demands treading a lonely path. First and foremost, I would like to express my sincere gratitude to my advisor Prof. Cristian Ciobanu for the continuous support to my Ph.D. study. I am grateful to him for allowing me to work with him, guiding me throughout and sharing his knowledge and experience with me, which has made my work an enjoyable experience. Even for once, I do not remember him getting impatient or angry over frustrating mistakes or missing points in my studies. He always encouraged me by praising my work, always preserved confidence in my ability to complete this Ph.D. He always gave me useful suggestions so that I could improve my research and writing skills. This experience has been extremely valuable to me and I will treasure it for the rest of my life. I am grateful to my co-advisor Prof. Vladan Stevanovićfor his scientific advice and knowledge and many insightful discussions and suggestions. I have learned a lot from working with him and interactions with him have always been lively. He was always a source of encouragement during demanding times. I will always be indebted to him for teaching me how to become a researcher, starting from technical issues and ranging to the organization of work and attention to detail. Besides my advisers, I would like to thank Prof. Geoff Brennecka, who was always there to listen to me and to give me sincere advice. He was always there to meet with me and talk about my ideas, and to ask me good questions to help me think through my problems. His enthusiasm is contagious and went a long way in developing my interest and understanding of the subject. I would like to thank my thesis committee: Prof. John Berger and Prof. Aaron Stebner, for their time and service on the committee also for their insightful comments which incented me to widen my research from various perspectives.

xvii I gratefully acknowledge the financial support provided by the National Science Foun- dation through Grant No. DMREF1534503. I would also like to thank the Golden Energy Computing Organization at Colorado School of Mines and the National Renewable Energy Laboratory for providing supercomputing resources. I would like to thank all fellow group member of Ciobanu/Stevanovićresearch group- Abdul, Anuj, Eric, and Félix-for keeping my mind refreshed which helped me to concentrate in work over and over again. I would like to thank Prashun for helping me in understanding pylada and developing codes without getting tired at my endless questions. My sincere thanks also go to Dr. Kappes for insightful discussions on atomistic modeling, algorithms, and coding languages. I had vibrant discussions with Badri and Tamoghna which have enriched my work. Kevin and Dong filled me with valuable discussions which went a long way in shaping this project. Special thanks to Abdul who freely and regularly provided me with personal guidance and his individual insights. I sincerely thank Tanaji who listened and discussed ideas about my research with me on many occasions. I would like to thank Diptak who helped with proofreading and other technical problems I encountered along the way. Above all, I must thank my parents who always stood by me in good times and bad. I love them so much, and I would not have made it this far without them. Thanks for everything that helped me get to this day.

xviii To my parents

xix CHAPTER 1 INTRODUCTION

Piezoelectric devices have played a crucial role in enabling technology for a vast range of industrial and consumer products [1, 2, 3, 4]. As published in the new market research report “Piezoelectric Devices Market-Industry Trends, Opportunities and Forecasts to 2023”, the piezoelectric device market is expected to witness a compound annual growth rate (CAGR) of 4.35% to reach a total market size of US$ 24.9 billion by 2023, from US$ 20.1 billion in 2018. Material, product, and application are considered as three major pillars of the piezoelectric device market [3, 4]. The material division is further subdivided into piezocrystals, piezoceramics, piezopolymers, and piezocomposites [5]. On the basis of product, the piezoelectric device market is subdivided into sensors, actuators, transducers, motors, and generators [3]. The application of piezoelectric devices is extensive in manufacturing aerospace, defense, automotive, healthcare, and consumer electronics segment [3, 6, 7]. In spite of its vast applicability, only a handful of piezoelectric materials are technologically relevant and cover all piezoelectric devices in microelectromechanical (MEMS) applications. The design and discovery of new piezoelectric materials is key to ﬁnding solutions to drive the growth of piezoelectric device market. These solutions demand superior properties and performance in the new devices, and have been the focus of research over the past decade [8, 9]. While standard engineering techniques are successful in meeting these challenges, such techniques are mostly experimental. As a result, the laborious identiﬁcation of individual materials with improved performance is inherently slow. It is possible to accelerate the discovery of these materials by understanding the processes that control material properties at atomistic level. This enables the tailoring of the structural and mechanistic features that lead to the improvement of the piezoelectric properties.

1 Density functional theory (DFT) is one of the most viable atomistic simulation technique in this regard. In the expansive utilization of ab initio techniques for the identification and design of new materials, DFT is particularly fast and accurate [10, 11, 12, 13]. Design of materials driven by computational techniques is heading the thrust to direct and guide syntheses of new piezoelectric materials. DFT technique is an ideal method for investigating atomistic phenomena in these materials. Additionally, this technique can also be utilized to investigate the response of these materials to external parameters such as pressure and lattice strain. The central theme of this thesis is to utilize DFT calculations to develop novel design strategies to enhance the properties of existing piezoelectric materials such as AlN. These strategies aim to enhance the functional properties, such as piezoelectric response using alloying, co-alloying, and external parameters such as strain. Subsequently, a high-throughput DFT screening technique is employed to aid the discovery of new piezoelectric materials. The conclusions from this work harbor the potential for the discovery of new piezoelectric materials with large piezoelectric response. Efficient combination of theoretical predictions supported with experimental confirmation will help the development of these materials with improved functionality, thus accelerating the discovery of novel and functionally advanced piezoelectric materials.

1.1 Piezoelectric Eﬀect and Applications

Piezoelectric materials are a special class of materials that generate a voltage under the application of stress. Since this effect is also observed in a reverse manner, a voltage across the sample will produce strain (or deformation) in the sample. The schematic representation of direct and converse piezoelectric effects are shown in Figure 1.1. This particular property is used in wide range of microelectromechanical devices [6, 14]. Piezoelectricity is observed in crystals having no center of symmetry, i.e., this effect does not occur in centrosymmetric materials. In centrosymmetric materials, there is no way for mechanical stress to generate a dipole moment as any motion of an ion in centrosymmetric

2 (a) (b) Applied Applied compressive tension (") stress (σ) + + Developed Developed charge charge - -

Developed Applied Developed strain electric field Applied strain electric field + - ∆$ - ∆$ +

Figure 1.1: Schematics of direct piezoelectric effects: (a) at applied compressive stress, (b) at applied tension. (c) and (d) describe the schematics of converse piezoelectric effect at applied electric field.

materials is reﬂected by symmetry to an equal and opposite motion that cancels out the dipole moment.

1.2 Classiﬁcation of Piezoelectric Materials

One subgroup of piezoelectric materials show pyroelectric behavior. Pyroelectricity refers to the change in polarization by the changes in structure due to thermal eﬀects [15]. In some of these pyroelectric materials, the polarization along the polar axis can be reversed by reversing the polarity of the electric ﬁeld. Such crystals are called ferroelectric, i.e., these are spontaneously polarized materials with reversible polarization. The special characters of ferroelectric materials are [16, 17]: (1) hysteresis behavior (P E curve), (2) spontaneous ∼ polarization, (3) reversibility of polarization, (4) ferro-paraelectric transition temperature or

Curie temperature (TC ). Above TC , the ferroelectric material transforms to a high symmetry paraelectric phase, where all piezoelectric activity disappears. Below TC , it shows a ferroelectric behavior and there is a ferroelectric to paraelectric transition at TC . The classiﬁcation

3 Dielectric

Piezoelectric

Pyroelectric

Ferroelectric

Figure 1.2: Classiﬁcation of piezoelectric materials. They are classiﬁed into of piezoelectric, pyroelectric, and ferroelectric materials. of piezoelectric materials are shown in Figure 1.2 and described in more detail in Appendix A.

1.3 Vector/Tensor Quantities Related to Piezoelectricity

Piezoelectricity is the combination of (a) the materials’ elastic behavior, (i.e.obey Hooke’s law) and (b) the materials’ electrical behavior. In this section, we will introduce several vector/tensor quantities related to these two behaviors. Vectors related to electrical properties: From the electrical properties of material perspective, we are concerned with the following three vectors: (1) E or the electric field intensity or field strength; equal to the force that would be exerted on a small unit charge placed at a point. (2) P or the polarization; the electric dipole moment per unit volume. (3) D or the electric displacement; defined by the charge per unit area that would be displaced across a layer of conductor placed across an electric field. Tensors related to mechanical properties:

4 Strain, ǫ is the response of a system to an applied stress σ. In matrix notation, the symmetric 2nd-rank strain tensor is given as,

ǫ11 ǫ12 ǫ13 ǫ = ǫ21 ǫ22 ǫ23 (1.1) ǫ31 ǫ32 ǫ33   The stress tensor σ, similar in nature to the strain tensor-both are symmetric second- order tensors, is given in matrix form as,

σ11 σ12 σ13 σ = σ21 σ22 σ23 (1.2) σ31 σ32 σ33   The Voigt notation or Voigt form is used to utilize the symmetry of condensed matter to transform second-order tensors (such as strain tensor and stress tensor) to vectors [18]. In Voigt notation ǫ is simpliﬁed to a 6-dimensional vector:

ǫ11 ǫ1 ǫ22 ǫ2 ǫ ǫ ǫ  33  3 (1.3) ≡ ǫ23 ≡ ǫ4     ǫ13 ǫ5     ǫ12 ǫ6     Similarly, representation in Voigt notation of σ is,

σ11 σ1 σ22 σ2 σ σ σ  33  3 (1.4) ≡ σ23 ≡ σ4     σ13 σ5     σ12 σ6     1.4 Constitutive Equations

The constitutive equations for linear piezoelectric materials are well documented in the IEEE Standard on Piezoelectricity [19]. Here, we briefly describe the constitutive relations among different variables involved in piezoelectric effect and define various coupling coefficients that connect these variables. Dielectric permittivity:

5 The induced polarization P in a dielectric material by an applied electric ﬁeld vector E, except for very high electric ﬁeld, is expressed as,

P = ε0χE (1.5)

where, X is the dielectric susceptibility of the material and χ is a second-rank tensor. ε0 = 8.854 10−12 F/m is the dielectric permittivity of vacuum. The vectors P, E and D are × connected by the following relation [20],

D = ε0E + P (1.6)

Combining eqs. (1.5) and (1.6),

D = ε0E + ε0χE = ε0(1 + χ)E = εE (1.7)

where, ε = ε0(1 + χ) is the dielectric permittivity of the material. In practice, the relative

r dielectric permittivity, ε = ε/ε0, also called the dielectric constant of the material, is more often used than the dielectric permittivity [15]. Elastic stiﬀness tensor: The relationship between the applied stress σ and resulting strain ǫ in a linear elastic material is given by Hooke’s law,

σ = Cǫ (1.8)

And C deﬁnes the elastic stiﬀness tensor. The inverse relation expressed as,

ǫ = Sσ (1.9) where, S is the elastic compliance tensor and related to C deﬁned by,

SC = CS = I (1.10)

Where, I is an identity matrix. Piezoelectric strain modulus and piezoelectric stress coeﬃcients: The diﬀerential form of eqn. (1.6),

dD = ǫ0dE + dP (1.11)

6 Therefore, if the electric ﬁeld in the crystal is held constant, dP = dD. The linear relationship between stress σ applied to a piezoelectric material and P is written as,

P = dσ (1.12) where, d is known as piezoelectric strain modulus or piezoelectric modulus. Whereas, piezoelectric stress coeﬃcients or piezoelectric coeﬃcient tensor, e which relate P and strain ǫ expressed by,

P = eǫ (1.13)

Then, the relation between e and d can be obtained straightforwardly from the strain- stress relation, i.e., eqn. (1.8). The relation between these two piezoelectric constants is simply,

e = dC (1.14) d = eS (1.15)

1.5 Figures of Merit of Piezoelectric Devices

Piezoelectric materials find wide variety of applications in MEMS devices. Each type of MEMS device has their own figures of merit (FOMs) that characterizes the performance of the corresponding devices. The various type of FOMs associated with different piezoelectric devices have been reviewed in this section.

1.5.1 Figures of Merit

There are ﬁve FOMs in piezoelectric devices that dictate the eﬀectiveness of the piezoelectric device. They are described below: Piezoelectric strain modulus or d: The absolute magnitude of the piezoelectric strain modulus which is expressed in eqn. (1.12). This is one of the important FOM in most of the piezoelectric devices particularly in actuators [6].

7 Piezoelectric voltage constant or g : The piezoelectric voltage constant related to piezoelectric sensor devices is expressed by the following equation [6], d g = (1.16) εε0

where, d is piezoelectric strain modulus, ε is dielectric constant, and ǫ0 is the permittivity of free space. Electromechanical coupling factor or k: The electromechanical coupling factor dictates the performance of piezoelectric devices in energy conversion. This is written as [6], Stored mechanical energy k2 = (1.17) Input electrical energy or, Stored electrical energy k2 = (1.18) Input mechanical energy

Energy transmission coefficient or λmax: All the stored energy cannot be used. The mechanical load defines the actual work done. This means that the systems with zero mechanical load or a complete clamp (no strain), no output work is done. The energy transmission coefficient is defined by [14], Output mechanical energy λ = (1.19) max Input electrical energy

Mechanical quality factor or QM : The mechanical quality factor (QM ) of a piezoelectric

−1 devices characterizes the sharpness of the electromechanical resonance spectrum. (QM ) is

also equal to the mechanical loss (tan δm). The QM value is very important in evaluating

the magnitude of the resonant displacement and strain. The term tan δm describes the heat in the piezoelectric sample operated at its resonance mode [14]. Acoustic impedance or Z: The acoustic impedance (Z) is a important FOM of piezoelectric devices which is used for evaluating the acoustic energy transfer between two materials

8 [6]. It is deﬁned as, pressure Z2 = (1.20) volume velocity · In solid materials,

Z = ρC (1.21) p where ρ is the density and C is the elastic stiﬀness of the material. Apart from above mentioned FOMs, the value of the dielectric constant also has an important role on the electrical matching in all piezoelectric devices.

1.5.2 Piezoelectric Devices

The wide variety piezoelectric devices can be classiﬁed into the following areas: (a) Sen- sors, (b) Actuators, (c) Ultrasonic transducers, (d) Resonators and Filters, (e) Transformers, and (f) Energy harvesters. Their description, applications, and FOMs are described below: Sensors: A piezoelectric sensor is a versatile tool that ﬁnds applications in quality assur- ance, process control, and for research and development in many industries [21, 22]. The sensitivity of a piezoelectric sensor needs to be adequately high such that the generated signal can be detected above the background noise [23, 24]. The sensitivity of the piezoelectric sensor is maximized when the piezoelectric voltage constant (g) is maximized [22]. Hence, an important FOM of a piezoelectric sensor is the ‘g’ constant. In practice, a large dielectric constant (or capacitance) is also desirable for sensors in order to overcome the losses associated with the electrical circuit. Unfortunately, an increase in ǫ results in a lower g constant, as described in 1.5.1. Another important FOM of the piezoelectric sensor is the electromechanical coupling factor k. The sensitivity of a piezoelectric sensor increases with increasing value of k. Currently, two main groups of piezoelectric materials are used for piezoelectric sensor applications: (a) polycrystalline piezoceramics (such as PZT-based [25]) and (b) single crystalline piezoceramics (such as gallium phosphate, quartz, and tourmaline). Usually, PZT- based piezoceramics have roughly two orders higher piezoelectric sensitivity compared to

9 those natural single crystal piezoelectric materials [21]. On the other hand, the less-sensitive

single-crystal materials have high and long term temperature stability and high QM [21]. Actuators: In converse piezoelectric effect, the piezoelectric materials expand or contract (deform) when an electrical field is applied. They generate mechanical motion and force on the application of electric field. Piezoelectric actuators (often called motors) harness this mechanical motion and provide very short strokes with high-frequency and fast response times [26]. The material requirements for these classes of devices are somewhat different from sensors applications. For instance, the piezoelectric materials for ultrasonic motors require a very hard piezoelectric with a high QM to suppress heat generation. Likewise, in pulse-drive motors a low permittivity piezoelectric materials is used for its small hysteresis [27]. Ultrasonic transducers: In ultrasonic transducers, the AC voltage makes piezoelectric- crystal oscillate at the same frequency and produce ultrasonic sound [28, 6]. In these devices, two of the most important FOMs are the electromechanical coupling factor of the main vibration mode and the acoustic impedance. Another materials parameter dielectric constant also has an important role on the electrical matching [7, 6]. For low and medium ultrasonic frequencies (between 20 Hz and a few hundred kHz), a high values of permittivity are required in obtaining sufficient practical values for the electrical impedance. The high values of elastic compliance are favorable to maintain the dimensionality of the piezoelectric crystal. The linear dimension governs the resonance of the piezoelectric element. However, the higher the frequency devices, the less convenient are these high values. In pulse-echo transducers, the use of short ultrasonic pulses requires a mechanical QM as low as possible for broad bandwidth. The piezoceramics present the widest variety in their elastic, electric and piezoelectric constants and, in general terms they have the most favorable properties for ultrasonic transducers. The choice of a piezoelectric material depends on the specific application to which the transducer will be destined.

10 Resonators and Filters: The piezoelectric resonators are used to either generate waves of specific frequencies whereas the filters are employed to remove unwanted frequency components from the signal, to enhance wanted ones or both [29]. They find applications in cell phones, global positioning system (GPS), wireless local area networks (WLAN), and bluetooth. The high dimensional stability (i.e., piezoelectric materials with high mechanical stability or high stiffness) and the low-temperature coefficient is required to keep resonant frequency constant [30, 31]. To select a piezoelectric material for these applications, several parameters must be taken into account such as electromechanical coupling factor k, mechan-

ical quality factor QM , chemical composition, and film deposition temperature [32]. The last two material parameters are essential for integration of filters above integrated circuit (IC) configuration [33]. The piezoelectric resonators, commonly made from quartz, are used as frequency refer-

ences. Piezoelectric materials such as PZT, quartz, GaPO4, and AlN are used in commercial ﬁlters [6]. Each material exhibits advantages and drawbacks. However, AlN provides the best trade-oﬀ in terms of electrical and mechanical properties. This is used in microelectronics technologies and for above IC component [34]. It provides the best balance of performance, CMOS compatibility, manufacturability, and reliability [34, 35].

Table 1.1: Piezoelectric devices and relevant FOMs. Each type of piezoelectric device has its own FOMs which characterize their performances.

Ultrasonic Resonators & Energy FOM Sensors Actuators transducers Filters Transformer Harvesters

εij ❉ ❉ ❉ ❉ d ❉ ❉ ❉ ❉ ❉ g ❉ ❉ ❉ ❉ k/k2 ❉ ❉ ❉ ❉ ❉ ❉

λmax ❉

QM ❉ ❉ ❉ ❉ ❉ ❉ Z ❉ ❉

11 Transformers: The operation principle of a piezoelectric transformer is a combined function of actuators and sensors so that energy can be transformed from electrical form to electrical form via mechanical vibration [36, 3]. Hence, the combined ﬁgure of merits of sensors and actuators are required for the piezoelectric transformer materials.

The materials used for piezoelectric transformers includes PZT-series and LiNbO3 [36, 3]. Most of the high-voltage piezoelectric transformers are made of PZT material [37, 36]. The

newly developed thickness extensional mode piezoelectric transformer is made of PbTiO3 and is very eﬃcient at high frequencies [36]. Energy harvesters: Vibration-based energy harvesting has received growing attention over the last decade. There are three major steps associated with piezoelectric energy harvesting as stated in the words of Uchino [6] :

• “Mechanical-mechanical energy transfer: This includes mechanical stability of the piezoelectric transducer under large stresses, and mechanical impedance matching”.

• “Mechanical-electrical energy transduction: This relates to the electromechanical coupling factor in the composite transducer structure”.

• “Electrical-electrical energy transfer: This includes electrical impedance matching, a d.c./d.c. converter to accumulate the energy into a capacitor or a rechargeable battery”.

Electromechanical coupling factor and the product of g and d are the most widely used FOMs in piezoelectric energy harvesters for evaluating the energy conversion efficiency of the harvester devices [38]. Cantilever geometric structure also plays an important role in enhancing the performance of the harvester efficiency. The rectangular shaped cantilever structure are most widely used MEMS based piezoelectric harvesters. AlN and PZT based thin films are widely used piezoelectric materials for energy harvesters. However, the output power of AlN-harvester is limited to low power levels due to the low piezoelectric modulus of AlN. On the other hand, the efficiency of PZT based energy harvester is low as they cannot withstand large amount of strain due to the low stiffness modulus of PZT [38].

12 (a) (b)

50 1- 3 composites PMN : Lead metaniobate 60 ) PZT (type 800)

(%) 50 PSC PVDF : Polymer t

k P(VDF-TrFE) : Copolymer 40 PMN PT BT : Bismuth titanate 30 P(VDF-TrFE) PSC : Piezoelectric single crystals 20 PVDF BT 1-3 composites : PZT and 10 polymer. 0 0 10 20 30 40 0 10 Z 20(MRa) 30 40

Figure 1.3: Correlation between diﬀerent FOMs (a) Mechanical quality factor (Q or QM ) vs. electromechanical coupling factor (k), (b) Electromechanical coupling factor vs. dielectric constant (ǫ), and (c) Thickness mode coupling factor (kt) vs acoustic impedance (Z). Reproduced from Ref. [39, 40, 41], with the permissions from Springer and Wiley.

In summary, a wide variety of FOMs and material parameters have been used to compare the performance of piezoelectric materials on diverse types of piezoelectric devices. This is shown in Table 1.1. It is very difficult to define a universal figure of merit for a large range of configurations or applications. For instance, in actuating and sensing, maximizing ‘d’ and ‘g’-constant is the goal to improve their performance. Likewise, ‘k′ for energy harvester devices quantify the ability in energy conversion. In similar fashion, high ‘QM ’ and high ‘k’ are desired to meet the demands of the advanced high power resonator devices. However, piezoelectric materials with high ‘QM ’ and high ‘k’ are incompatible (shown in Figure 1.3-

13 a). Similarly, high dielectric constant of a sensor materials is desirable to over-come the losses associated with the electrical circuit. Despite the fact that no material co-exhibit

high dielectric constant and high ‘QM ’ (as shown in Figure 1.3-b). It can be observed that no material allows to obtain both high coupling factor and low acoustic impedance (Z) (as shown in Figure 1.3-c). High k-constant is necessary to enhance the sensitivity of the medical imaging transducers whereas, another FOM ‘Z’ should be as low as possible to match with the acoustic impedance of biological tissues. Thus, a compromise must be made depending on the target devices. In addition, a high TC is always desired for improved temperature stability for high temperature piezoelectric devices [42] and high QM for dimensional stability and low mechanical loss. All these explain the complexity of optimizing piezoelectric properties in existing piezoelectric materials. In chapter 3, we develop novel alloying strategies, and it reveals that co-alloying with YN and BN enables the most superior combination of piezoelectric and mechanical properties in AlN.

1.6 Piezoelectric Materials: Present Status and Motivation for Computational Investigations

This section describes the state-of-the-art reviews of modern piezoelectric materials, primary the materials based on ceramics (piezoceramics). The piezoceramics are the most widely used and commercially accepted piezoelectric material. They display large displacements or induce large electric voltages. Furthermore, they are cost-effective and can easily be tailored to meet the requirements of specific applications. Therefore, piezoceramics are the most preferred material for manufacturing piezoelectric devices and occupy the largest size of the overall piezoelectric device market [4]. Piezoceramics are broadly categorized based on their form of use, such as materials used in single crystal, polycrystalline, and thin film form.

14 1.6.1 Materials based on Single Crystal

Quartz [43, 44, 45] and some other piezoelectric crystals such as LiNbO3 [45] or LiTaO3

[45], GaPO4 [46] and more recently crystals from the Langasite [45] family are used in their single crystal form. These materials ﬁnd applications as raw materials to build resonators, ﬁlters, oscillator, and sensors [47]. This section reviews the application, properties, and limitations of various piezoelectric single crystal materials (Figure 1.4).

Figure 1.4: The advancement of piezoelectric single crystals with year. Schematic is sourced from Ref. [48]

15 Quartz: Quartz is one of the earliest piezoelectric materials used as timers in clocks and is also used for various acoustic devices, voltage controlled oscillators, and micro biosensors [43, 44, 45, 49]. In Figure 1.5 schematics of diﬀerent internal components of a watch (including quartz oscillator) are shown. The dielectric constant and piezoelectric modulus of single crystal quartz were found to be 4.5 and 2 pC/N [50], respectively. Piezoelectric α-quartz has a phase transition temperature of 573°C(α-quartz β-quartz). However, its maximum use temperature is normally limited → to 350°C. Beyond 350°C crystal structure in quartz experiences twinning, that causes the piezoelectric properties of quartz to disappear [45, 17].

The main advantage of quartz crystal is its extremely high mechanical quality factor QM (=106) [49]. This property of quartz crystal beﬁtting for applications demanding very high mechanical stability. Another important feature of quartz is its high abundance in nature. As a result, quartz crystals are available at low cost, and with relatively high purity and perfection. However, their applications in large strain applications are often limited due to

its low piezoelectric modulus (compared to AlN, PZT-based, BaTiO3-based), low coupling

constant (compared to PZT-based, BaTiO3-based), low voltage constant (compared to PZT-

based, BaTiO3-based), and low phase transition temperature (compared to AlN, langasite- based piezoelectrics).

Lithium niobate/tantalate: Lithium niobate (LiNbO3 or LN) and lithium tantalate (LiTaO3 or LT) crystals belong to an isomorphous crystal system and are structurally composed of oxygen octahedrons. Thus, both materials hold very important positions in the SAW device application ﬁeld. The Curie temperatures of LiNbO3 and LiTaO3 are 1210 and 660°C, respectively [52]. Piezoelectric single crystals such as 128°Y-X (128 °-rotated-Y-cut and X-

propagation)-LiNbO3 and X-112 °Y (X-cut and 112°-rotated-Y-propagation)-LiTaO3 have been extensively used as SAW substrates for applications in VIF ﬁlters [53, 54].

The piezoelectric modulus of LiNbO3 and LiTaO3 are 8.3 and -21 pC/N, respectively [52, 54]. These materials have high electromechanical coupling coeﬃcients compared to

16 battery quartz oscillator microprocessor circuit board coil magnet

Figure 1.5: α-Quartz used as a piezoelectric oscillator in a watch. Schematic is sourced from Ref. [51] quartz for surface acoustic waves. In addition, large single crystals can easily be obtained from their melt using the conventional Czochralski technique [55]. However, they have less mechanical quality factor compared to quartz [42]. The application temperature range for the LN crystals is limited to 650°C due to their low resistivity [53].

Langasite-based piezoelectrics: Langasite (R3Ga5SiO14, R= rare earth element) crystals have recently become attractive for high-temperature bulk acoustic wave (BAW) and surface

acoustic wave (SAW) applications [56, 57]. La3Ga5SiO14 or LGS has the highest piezoelectric modulus among this family of crystals [58]. Contrary to quartz, LN, and LT piezoelectric crystal, LGS has no phase transition up to its melting point at 1475°C [42]. Therefore, it becomes a suitable candidate for high-temperature applications in sensors. They can be conventionally grown by the Czochralski technique [59]. The dielectric constants and piezoelectric coeﬃcients of LGS are found to be in the range

of 18-25 and 5-8 pC/N, respectively [57]. The piezoelectric modulus d11 of LGS is about 2.7 times larger than quartz. This allows realizing piezoelectric sensors with langasite family of crystals signiﬁcantly with higher sensitivity. The electromechanical coupling factor is 2.3-3 times larger than quartz. As a result, the resonators with high pullability can be

17 fabricated with LGS piezoelectrics, having temperature stability with broad temperature and a high Q-factor. The Q-factors of Langasite piezoelectric crystals are in the same order of magnitude as those of quartz resonators. Langasite family is well suited for operating at high temperatures and mechanical loads as wireless acoustic and piezoelectric sensors.

GaPO4 piezoelectrics: Gallium orthophosphate is a quartz-analogue crystal. Currently,

GaPO4 with its properties is by far the best suited piezoelectric material to be used in sensor applications for machine monitoring and pressure measurements [60]. SAW micro-devices based on GaPO4 operate at temperatures of up to 800°C [60]. It possess a high d11 value of 4.5

pC/N and a low dielectric constant of 6 [46, 61]. In comparison with quartz, GaPO4 exhibits nearly all the advantages of quartz, additionally it also possess higher electromechanical coupling and has thermally stable physical properties up to 950 °C [61, 62]. At 950°C,

an irreversible phase transformation occurs and GaPO4 transforms into a cristobalite-like phase [42]. They can be grown using hydrothermal techniques and presently are produced commercially in autoclaves [63]. Single crystal relaxor ferroelectrics with the morphotropic phase boundary (MPB) composition show tremendous promise as ultrasonic transducers and electromechanical actuators

[64, 65]. The single crystals of Pb(Mg1/3Nb2/3)O3 (PMN), Pb(Zn1/3Nb2/3)O3(PZN) and their

binary systems with PbTiO3 (PMN-PT and PZN-PT) have extremely large electromechanical coupling factors [66, 67, 65]. PZN-8%PT single crystals are found to possess a high k33 value of 0.94 for the (001) crystal cuts [65]. This is much higher compared to the k33 of conventional PZT ceramics of around 0.70-0.80 [67, 65]. The single crystal of KNbO3 (KN) is known to have high piezoelectric activities [68, 69]. However, it is diﬃcult to obtain a dense ceramic body of KN by the ordinary ﬁring process [70].

1.6.2 Materials Based on Polycrystalline Materials

Polycrystalline materials in bulk form (ceramics and thick ﬁlms) with randomly oriented grains must be ferroelectric for exhibiting piezoelectricity. By the process called ‘poling’, the randomly distributed ferroelectric domains orients on the direction of electric ﬁeld [17].

18 Contrary to bulk samples, the thin film samples allow the controlled growth of thin layers (on the order of 0.1 to 1.55 µm) of many materials [71]. In case of non-ferroelectric polycrystalline films, to achieve piezoelectricity, one should orient the grains in a direction which is piezoelectrically active. Many non-ferroelectric materials which cannot be grown as single crystals and which cannot show piezoelectricity in the bulk form can thus be used in piezoelectric applications as thin films. This has significant technological consequences because thin layers may be deposited on different substrates and can be microstructured and integrated with other electronic or mechanical components [72]. The polycrystalline piezoelectric materials are broadly classified into following groups: (a) perovskite-based, (b) non-perovskite based, and (c) thin-film based. The following sections summarize the applications, properties, and limitations of several commercial polycrystalline piezoelectric materials. (a) Perovskite-based

The well-known piezoelectric perovskite materials are BaTiO3 (BT), PbTiO3, KNbO3, and mixed-cation compounds such as Pb(Mg1/3Nb2/3)O3 [17, 23, 73]. Perovskites easily form solid solutions with other perovskites, and hundreds of compounds have been synthesized with very interesting dielectric, optical, and piezoelectric properties [7, 6]. The chemically simple perovskite materials may undergo a complex sequence of phase transitions. Some materials, like BaTiO3, KNbO3, and NaNbO3, undergo several phase transitions into successive ferroelectric phases, whereas, others like relaxor-ferroelectric solid solutions, may exhibit a relaxor phase over a narrow temperature range before transforming into a ferroelectric phase [74]. In Na1/2Bi1/2TiO3 based compositions, a phase transition sequence from a paraelectric to antiferroelectric relaxor to ferroelectric phase has been observed [75]. In general, the piezoelectric coeﬃcients in ferroelectrics are higher than in nonferro- electrics and in perovskites range from tens to hundreds of pC [7, 71, 76]. Besides this high piezoelectric activity, an advantage of ferroelectric materials over nonferroelectric ones is that

19 they can be prepared in polycrystalline form (as bulk ceramics). They exhibit piezoelectric effect after poling with a strong electric field [77]. As ceramics are usually easier and less expensive to prepare than single crystals, ferroelectric perovskites have been widely used for various piezoelectric applications, almost exclusively in polycrystalline form [71, 78]. PZT-based: Among polycrystalline piezoceramic materials, by far the most important piezoelectric materials are solid solutions of PbZrO3 and PbTiO3 (Pb(Zr, Ti)O3 or PZT) which have been used in hundreds of different devices [79]. For a long time, PZT was considered as the best piezoelectric material and main research efforts have been directed toward optimization of PZT based compositions for specific applications. Although this material has a high piezoelectric response, 410 pC/N [76], it is limited by its maximum operating temperature of 250°C. Recently, however, it has been shown that the piezoelectric properties superior to those in PZT ceramics may be obtained in single crystals of some perovskites [80]. However, considering the toxicity of lead oxide, development of lead-free piezoceramics is encouraged in recent years.

BaTiO3(BT)-based: Barium titanate was the ﬁrst piezoelectric ceramic developed and is still widely used [7]. This is because with the exception of Rochelle salt, its coupling factors are substantially higher than that of any previously known material [81, 31]. These ceramics

have a relatively high electromechanical coupling factor, k33 (0.50), and piezoelectric strain

constant, d33 (190 pC/N), and have been used in piezoelectric applications such as sonars [82, 81]. In addition, they are chemically more stable than Rochelle salt, have a wider temperature range of operation, and superior manufacturability.

The dielectric and piezoelectric properties of ferroelectric ceramic BaTiO3 can be aﬀected

by its own stoichiometry and microstructure. Modiﬁed ceramic BaTiO3 with dopants such as Pb or Ca ions have been developed to stabilize the tetragonal phase over a wider temperature range and have been used as commercial piezoelectric materials. After the discovery of PZT, BT’s role in piezoelectric devices faded out and they are primarily used as capacitors at

present days[17]. Other major disadvantages of BaTiO3-based piezoelectrics is its low Curie

20 temperature. Thus, the working temperature range of this ceramic is narrow for actual piezoelectric applications [83].

KNbO3-based: Solid solutions of potassium niobate (KNbO3) with sodium niobate (NaNbO3)

leads to a system with many MPBs, showing ferroelectricity up to about 90% NaNbO3 [84]. There is a MPB near the 50/50 composition separating two orthorhombic phases. This composition shows a rather high piezoelectric longitudinal response, a transverse coupling

coeﬃcient between those in modiﬁed lead titanate and hard PZT(d33 reported to be as high

as 160 pC/N for (K0.5Na0.5NbO3 (KNN)), and a relatively low dielectric constant (around 400 at room temperature) [84, 85, 86]. KNN is one of the most lead-free candidates for high-frequency transducers, with its low density, high Curie temperature, and relatively high electromechanical coupling factor [85]. They could, therefore, be considered as a viable alternative to PZTs for selected applications. In addition, this material in a form of fully dense ceramics appears to be compatible with animal and human tissues.

KNbO3 and K1−xNaxNbO3 ceramics are difficult to process to full densities by natural sintering for several reasons [83, 6]. One is that the phase stability is limited to 1040°C for KNbO3 and 1140°C for K1−xNaxNbO3. Therefore, the high sintering temperatures are not possible. In addition, slight changes in stoichiometry (both on the niobium rich and the potassium-rich side) lead to the formation of extra phases [83]. Potassium excess might provide some densification, helping liquid phase but some of the formed extra phases are highly hygroscopic and lead to a quick disintegration of the final sample once exposed to humidity. Some concern has been raised about high dielectric losses in these materials [83]. (b) Non-perovskite-based:

Bismuth Titanate Based Layered Compounds: The crystal structure of Bi4Ti3O12 (BIT) and related compositions, ﬁrst described by Aurivillus [87, 88], is characterized by pseudo- perovskite layers. They have a high ferroelectric Curie temperature (up to 980°C) and show high-temperature piezoelectric behavior. They are potential candidate lead-free materials in piezoelectric device applications, especially under high temperature and high-frequency

21 conditions [88]. They have relatively low dielectric constant, low aging rate, and strong anisotropic electromechanical coupling factors. Unfortunately, the piezoelectric activity if BITs is relatively low, with values less than 20 pC/N [88]. (c) Thin film-based: Both zinc oxide (ZnO) and aluminum nitride (AlN) are simple binary compounds with a Wurtzite-type structure, which can be sputter-deposited as a c-axis oriented thin film on a variety of substrates [89, 71]. Since ZnO has a reasonable piezoelectric coupling, thin films of this material are widely used in bulk acoustic and surface acoustic wave devices. The fabrication of highly oriented (along c-axis) ZnO films have been studied and developed extensively. However, the performance of ZnO devices is limited, due to their low piezoelectric coupling (20-30%) [89, 71]. PZT thin films are expected to exhibit higher piezoelectric properties. At present, the development of PZT thin films is being carried out for use in micro-transducers and micro- actuators [71]. AlN and ZnO exhibit much higher mechanical quality factors than PZT, and in comparison to LiNbO3, they are much more easily grown in thin film form. Moreover, integration and process compatibility with the rest of the device is less difficult using the relatively simple wurtzite materials. The strong polarity of their crystalline structure allows for a polar growth and a stable piezoelectric response with time, whereas ferroelectrics always risk depoling. AlN-based: It is a hard material with light atoms and exhibits a high mechanical quality up to very large microwave frequencies. Moreover, it is CMOS compatible, in contrast to PZT and ZnO. A rather specific feature of AlN is its high sound velocity. AlN exhibits quite a large bandgap of 6.1 eV at room temperature. Consequently, it is a good dielectric material up to high temperatures. PZT thin film solutions in micro-sensors and actuators are very competitive when current signal and force or power outputs are demanded. When voltage signal, intrinsic signal-to-noise ratio, or power efficiency is an issue, the nonferroelectric AlN is competitive, or even better [71].

22 1.6.3 Toxicity of Lead-based Piezoelectrics

Lead oxide based piezoelectric, which are commonly represented by lead zirconate titanate (Pb(Zr,Ti)O3) or PZT) are the most widely used materials for piezoelectric actuators, sensors and transducers due to their excellent piezoelectric properties [77, 17]. Nonetheless, there is a raising concern about the recycling and disposal of piezoelectric devices containing PZT due to the toxicity of lead oxide [90, 91]. Lead oxides vaporizes during processing and stays in the environment causing damage to brain and nervous system [92, 93]. This is particularly important in PZT containing consumer products such as cars, various kinds of smart systems, and sound generators [91]. Thus, there is an increasing tendency to look for alternative piezoelectric materials that does not contain Pb.

1.6.4 Motivation for Computational Research on Improving Piezoelectric Ma- terials

The piezoelectric properties and applications of various commercial piezoelectric materials have been extensively discussed in section 1.6. For convenience, the primary merits and demerits of such piezoelectric materials are summarized in Table 1.2. This infers that each commercial piezoelectric has their own drawbacks. Thus, it is necessary to improve the piezoelectric and related properties of existing materials, as well as discover new piezoelectric materials. Enhance the properties of existing piezoelectric materials would help to make them technologically more attractive. On the other hand, discovery of new piezoelectric materials will expand the pool of existing piezoelectric materials, which would broaden the range of their applications and could potentially replace the toxic materials presently in use. Over the past century, materials scientists have successfully engineered the functionality of materials by chemical substitution (cation and anion) or alloying. This technique leads to enormous possibilities to combine the elements of the periodic table into diﬀerent alloys if we allow any number of components at any concentrations. Manufacturing each of these materials and measuring the properties of each one of them would take an unbearably long time and require vast resources. Therefore, it is imperative that we realize how materials

23 Table 1.2: Merits and demerits of several commercially used piezoelectric materials

Materials Advantages Disadvantages Quartz (a) High mechanical quality factor (a) Low piezoelectric modulus (b) Temperature stability (b) Low coupling constant (c) High dimensional stability (c) Low phase transition temperature LiNbO3 & (a)HighCurietemperature (a)Lowmechanicalstability LiTaO3 (b)Non-toxic (b)CMOSincompatible Risk of depoling Langasite-based (a) No phase transition up (a) Twinning and softness (La3Ga5SiO14) toitsmeltingpoint ofthismaterialsmaylimit (b) High piezoelectric and its applications at high pressure electromechanical properties (c) Low acoustic wave propagation losses BaTiO3-based (a) High piezoelectric properties (a) Low mechanical stability (b) High Coupling Constants (b) Low Curie temperature (c)Non-toxic (c)Riskofdepoling PbTiO3-based (a) High piezoelectric properties (a) Low mechanical stability (b) High Coupling Constants (b) Low operational temperature (c) Highly toxic (d) CMOS incompatible (e) Risk of depoling KNbO3-based (a)Non-toxic (a)Lowpiezoelectricproperties (b) High Curie temperature (b) High dielectric losses (c) Bio-compatibility (c) High temperature synthesizability (d) Volatility of alkali components at higher temperature (e) Low sintered density; diﬃcult to fabricate the ceramics ZnO-based (a) High mechanical quality factor (a) Low piezoelectric modulus (b)Non-toxic (b)Lowcouplingconstant (c) CMOS incompatible AlN-based (a) High mechanical quality factor (a) Low piezoelectric modulus (b) Temperature stability (b) Low coupling constant (c) CMOS compatible

24 work at an atomic and electronic scale, and use that knowledge to guide our search for better materials with improved properties. There are plenty of advantages with computer simulations compared to experiments. It is possible to execute simulations with toxic or radioactive materials. It is also feasible to test costly materials without any cost. Although, there is also a constraint in computer simulations. There is often a choice to be made whether a simulation should be accurate or if it should be large (e.g. include many atoms or long time scales). This Ph.D. thesis is an effort to extend the frontiers of disordered multicomponent nitrides for efficient designing of nitride piezoelectrics and to discover novel piezoelectric materials for replacing non-toxic piezoelectrics. The first half of this work, we have explored the effect of alloying and co- alloying on improving the properties of AlN. Another half of this work, we have described our effort in finding new piezoelectrics by means of high-throughput DFT. Note that, materials properties can controlled by the processes that occur at other length scales such as macroscale [94, 95] and mesoscale [96, 97, 98, 99], which are not the focus of the present thesis.

1.6.5 Prior Computational Studies of Piezoelectric Properties

AlN-Based Piezoelectrics: Numerous eﬀorts were made over the years to predict piezoelectric properties of AlN by DFT. Kamiya [100] calculated the piezoelectric properties of AlN. The predicted d33 of AlN is 6.7 pC/N which is about 20% higher than the experimentally measured values. Later, Bernardini and Fiorentini [101] calculated d33 as 5.4 pC/N. The latter value is very close to the experimental value. Tasnadi et al. [102] published an

2 e33 value of 1.55 C/m , identical to the experimental value. With the discovery of improvement of piezoelectric constant by Sc doping, the eﬀorts are being done more systematically in order to predict Al1−xScxN properties. The most complete work was published by Caro et al. [103]. The values for e33, and d33 were obtained as 1.46 C/m2, and 5.12 pC/N respectively. All values are thus about 10% smaller than in reality. Recently, it was discovered that Al substitution by Sc allows for an increase of the piezoelectric response [103, 102] of AlN. The origins of this enhancement in piezoelectric response

25 have been studied [102]. As the content of the rocksalt end member in the alloy increases, the accompanying structural frustration enables a greater piezoelectric response. Although the peak of the piezoelectric increase is found at around 50% Sc, it becomes increasingly difficult to maintain good enough crystalline quality in the films to realize the exceptional piezoelectric response. Therefore, it is expected that practical device applications will be using lower concentrations of Sc. However, there are several other challenges involved in AlN based thin piezoelectric thin films.

• The possible route for improving its piezoelectric modulus is addition of transition metal element. Increasing piezoelectric modulus is also accompanied by elastic softening which make this material less appealing for resonator based devices where mechanical stability of the thin ﬁlm is one of the important ﬁgure of merit.

• The increasing alloy content results in severe degradation of thin film texture. The piezoelectric responses of piezoelectric thin film strongly depend on the texture of the thin film. So it is preferable to choose proper alloying with the highest piezoelectric sensitivity to alloy composition.

Computational Searches for New Piezoelectric Materials: Identifying new piezoelectric systems required high-throughput screening of inorganic materials for their piezoelectric properties. The need for new piezoelectrics was recognized before and was the main motivation behind recent computational eﬀorts in high-throughput screening of inorganic materials for piezoelectric performance. These include the work on perovskite alloys, and creation of a database of piezoelectric properties of compounds. Materials with reduced dimensionality, such as 2D mono- and multi-layers have also been investigated recently.

1.7 Objectives

My doctoral research work aims to discover novel piezoelectric materials and enhance the properties of existing (AlN) ones to make them technologically attractive.

26 1.7.1 Improving Performance of Existing Piezoelectric Materials

In section 1.6.5, we have identified that main drawback of AlN piezoceramics is its low piezoelectric modulus. Addition of transition metal elements is a suggested method to improve the piezoelectric modulus of AlN. However, such alloying severely degrades the thin film texture, ultimately deteriorating the piezoelectric properties. Thus, it becomes necessary to identify suitable alloying elements that may yield highest piezoelectric response with minimal additions. In my doctoral research, we approached the problem by conducting DFT calculations of structure and properties of AlN system alloyed with different transition elements such as Sc3+,Y3+, Cr3+, and La3+. The study revealed that alloying with Cr3+ would lead to the lowest wurtzite/rocksalt transition point, thereby allowing piezoelectric enhancements at alloying levels that are easier to stabilize during the synthesis. The significant enhancement of piezoelectric modulus, with minimal alloying additions, establish CrxAl1−xN as a suitable replacement for AlN [104]. However, such alloying is also accompanied by a softening of the AlN lattice. Our subsequent research focused to develop novel alloying strategies, and it was revealed that co- alloying with YN and BN enables superior combination of piezoelectric and mechanical properties in AlN [105].

1.7.2 Searching for New Piezoelectric Materials

Besides improving the existing piezoelectrics, we also used high-throughput DFT approach to discover new piezoelectrics. The main goal of this work is not to discover the single piezoelectric material that will replace all others, instead, expand the pool of existing piezoelectric materials with different set of other properties with the focus on large piezoelectric modulus. This would efficiently guide users towards new materials that will best fit their needs.

27 Our focus is to find new piezoelectric materials with large piezoelectric response i.e. materials with large d-value. Here, we have used the relationship between d and C as discussed in 1.14, where, we see that piezoelectric modulus d is inversely related to its C. This implies that the softer materials tend to have larger piezoelectric modulus compared to a stiffer materials with similar e. Even if materials with non-interesting e, because of softness one might get large d in the softer direction(s). This knowledge encourages us to inspect in the quasi-2D van der Waals (vdW) bonded layered materials for new piezoelectric materials. The quasi-2D materials exist in layers and the layers are held together by soft vdW bond. Hence, they are very soft in vdW-bonded direction. As a result, one may expect large d in that softer direction because of low stiffness in that direction. 51 vdW layered materials are predicted to have piezoelectric response larger than that of AlN and 3 materials with response even larger than PbTiO3. More importantly, we identified 33 materials with large piezoelectric response that do not contain toxic elements such as Pb. None of these predicted candidates have previously been considered as piezoelectric materials. Furthermore, our analysis shows that large piezoelectric response in layered materials is indeed enabled by the soft vdW bonds, i.e., large d originates from the direct shear or axial deformations of the vdW bonds. This research established a wide scope to synthesize quasi-2D materials for applications demanding high piezoelectric modulus.

1.8 Organization of the Thesis

This thesis is organized in the following chapters:

• Chapter 2 presents the work titled “Enhanced piezoelectric response of AlN via CrN alloying” published in Physical Review Applied. Here, we have studied the eﬀect of Cr alloying on piezoelectric and mechanical properties of AlN. The study revealed that alloying with CrN would lead to the lowest wurtzite/rocksalt transition point, thereby allowing piezoelectric enhancements at alloying levels that are easier to stabilize during the synthesis. The signiﬁcant enhancement of piezoelectric modulus, with minimal

28 alloying additions, establish (Cr,Al)N as a suitable replacement for AlN.

Author’s contribution: In this work, I took part in planning, performed all the theoretical calculations, participated in the evaluation and interpretation of the results, and was responsible for writing the manuscript.

• Chapter 3 describes the eﬀect of co-alloying on the piezoelectric and mechanical properties of AlN. My initial research established that the piezoelectric response of Alu- minum Nitride (AlN), a common piezoelectric for resonators, could be improved by alloying with trivalent transition elements like chromium Cr+3, yttrium Y+3, scandium Sc+3 and lanthanum La+3. However, such alloying is also accompanied by a softening of the AlN lattice. Our subsequent research focused to develop novel alloying strategies, and it was revealed that co-alloying with YN and BN enables the most superior combination of piezoelectric and mechanical properties to develop in AlN. This work is published Journal of Applied Physics titled “Tuning the piezoelectric and mechanical properties of the AlN system via alloying with YN and BN”.

• Chapter 4 describes the high throughput discovery of new piezoelectric materials using first principle density functional theory. The discovery of new piezoelectric materials would broaden the range of applications and offer cost-effective, non-toxic alternatives. In this work, we have performed a large-scale computational investigation of the piezoelectric properties of 869 vdW layered materials, including binary and ternary chemistries. The results confirm our expectation. Our search has revealed a number vdW layered materials that exhibit large piezoelectric response; 51 vdW layered materials are predicted to have piezoelectric response larger than that of AlN, a commonly used piezoelectric material in resonators and 3 materials with response even larger than

29 PbTiO3. More importantly, we identiﬁed 33 materials with large piezoelectric response that do not contain toxic elements such as Pb. None of these predicted candidates have previously been considered as piezoelectric materials. Furthermore, our analysis shows that large piezoelectric response in layered materials is indeed enabled by the soft vdW bonds, i.e., large d originates from the direct shear or axial deformations of the vdW bonds.

Author’s contribution: In this work, I took part in planning, performed majority the theoretical calculations, participated in the evaluation and interpretation of the results, and was responsible for writing the manuscript.

• Chapter 5 summarizes the work presented in this thesis, highlights the main conclusions and provides directions for future work.

30 CHAPTER 2 ENHANCED PIEZOELECTRIC RESPONSE OF AlN via CrN ALLOYING

A paper published in Physical Review Applied1 Sukriti Manna2, Kevin R Talley3, Prashun Gorai3, John Mangum3, Andriy Zakutayev4, Geoff L Brennecka3, Vladan Stevanović3, Cristian V Ciobanu2 Since AlN has emerged as an important piezoelectric material for a wide variety of applications, efforts have been made to increase its piezoelectric response via alloying with transition metals that can substitute for Al in the wurtzite lattice. Herein, we report density functional theory calculations of structure and properties of the Cr-AlN system for Cr concentrations ranging from zero to beyond the wurtzite-rocksalt transition point. By studying the different contributions to the longitudinal piezoelectric coefficient, we propose that the physical origin of the enhanced piezoelectricity in CrxAl1−xN alloys is the increase of the internal parameter u of the wurtzite structure upon substitution of Al with the larger Cr ions. Among a set of wurtzite-structured materials, we have found that CrxAl1−xN has the most sensitive piezoelectric coefficient with respect to alloying concentration. Based on these results, we propose that (Cr, Al)N is a viable piezoelectric material whose properties can be tuned via Cr composition. We support this proposal by combinatorial synthesis experiments, which show that Cr can be incorporated in the AlN lattice up to 30% before a detectable transition to rocksalt occurs. At this Cr content, the piezoelectric modulus d33 is approximately four times larger than that of pure AlN. This finding, combined with the relative ease of synthesis under non-equilibrium conditions, may propel (Cr, Al)N as a prime piezoelectric material for applications such as resonators and acoustic wave generators.

1Reprint of Physical Review Applied 9 (3), 034026 (2018) with permission from American Physical Society 2Department of Mechanical Engineering, Colorado School of Mines, Golden, CO 80401, USA 3Department of Metallurgical and Materials Engineering, Colorado School of Mines, Golden, CO 80401, USA 4National Renewable Energy Laboratory, Golden, CO 80401, USA

31 2.1 Introduction

Aluminum nitride has emerged as an important material for micro-electromechanical (MEMS) based systems [106, 107] such as surface and bulk acoustic resonators [106, 108], atomic force microscopy (AFM) cantilevers [106], accelerometers [109, 110], oscillators [111], resonators for energy harvesting [112, 113], and band-pass filters [114]. The advantages of using AlN in MEMS devices include metal oxide semiconductor (CMOS) compatibility, − − high thermal conductivity, and high temperature stability. In addition, its low permittivity and high mechanical stiffness are particularly important for resonator applications [106, 115]. However, the piezoelectric constants of AlN thin films are lower than those of other commonly used piezoelectric materials. For example, the out-of-plane piezoelectric strain modulus [116] d33 of reactively sputtered AlN films is reported to be 5.5 pC/N, whereas d33 for ZnO can be at least twice as large [117], and PZT films can be over 100 pC/N [107]. It is therefore desirable to find ways to increase the piezoelectric response of AlN in order to integrate AlN-based devices into existing and new systems. A common way to engineer piezoelectric properties of AlN is by alloying with transition metal nitrides (Sc, Y, others), which can lead to a several-fold increase in the field-induced strain via increases in the longitudinal piezoelectric coefficient e33 and simultaneous decreases in the longitudinal elastic stiffness C33 [76, 103, 105, 102]. In the case of ScN alloying, the origins of this response have been studied [102], and it is presumed that other such systems which also involve AlN alloyed with rocksalt-structured end members are similar:as the content of the rocksalt end member in the alloy increases, the accompanying structural frustration enables a greater piezoelectric response. This structural frustration, however, is also accompanied by thermodynamic driving forces for phase separation [118] which, with increased alloy concentration, lead to the destruction of the piezoelectric response upon transition to the (centrosymmetric, cubic) rocksalt structure. The experimental realization of large alloy contents without phase separation or severe degradation of film texture and crystalline quality can be quite difficult [118, 119], even when using non-equilibrium deposition processes such as sputtering.

32 Thus, it is desirable to ﬁnd alloy systems for which the structural transition from wurtzite to rocksalt occurs at low alloying concentrations since these may be more easily synthesized and more stable, while also (hypothetically) providing comparable property enhancements as those observed in the more-studied (Sc, Al)N alloy system. Among the AlN-based systems presently accessible experimentally, (Cr, Al)N has the lowest transition composition between the wurtzite and rocksalt structures, occurring at approximately 25% CrN concentration [120, 121]. This motivates the investigation of the piezoelectric properties of the

CrxAl1−xN system, which we also refer to, for simplicity, in terms of Cr substitution for Al. In this article, we study Cr-substituted AlN using density functional theory (DFT) calculations of structural, mechanical, and piezoelectric properties. Given that Cr has unpaired d electrons, a challenge to overcome in these calculations is the simulation of a truly representative random distribution of the spins of Cr ions, whose placement in the AlN lattice involves not only chemical disordering, but spin disordering as well. Among a set of wurtzite-based materials, we have found that Cr-doped AlN is the alloy whose piezoelectric stress coeﬃ- cient e33 is the most sensitive to alloying concentration and also has the lowest wurtzite- to-rocksalt transition composition. The key factor leading to the enhanced piezoelectricity in CrxAl1−xN alloys is the ionic contribution to the coeﬃcient e33; this ionic contribution is increased through the internal u parameter of the wurtzite structure when alloyed with the

(larger) Cr ions. Therefore, we propose CrxAl1−xN as a viable piezoelectric material with properties that can be tuned via Cr composition. To further support this proposal, we have performed combinatorial synthesis and subsequent characterization of CrxAl1−xN ﬁlms, and have showed that Cr can be incorporated in the AlN lattice up to 30% before a detectable transition to rocksalt occurs. At this Cr content, the piezoelectric modulus d33 is four times larger than that of AlN. Pending future device fabrication and accurate measurements of properties and device performance, this signiﬁcant increase in d33 can propel Cr-AlN to be the choice material for applications such as resonators, GHz telecommunications, or acoustic wave generators.

33 2.2 Methods 2.2.1 Paramagnetic Representation of Cr-AlN Alloys

Starting with a computational supercell of wurtzite AlN, any desired Cr concentration is realized by substituting a corresponding number of Al ions with Cr ions in the cation sublattice. In order to realistically simulate the chemical disorder of actual (Cr, Al)N alloys while maintaining a tractable size for the computational cell, we use special quasirandom structures (SQS) [122, 123, 124]. The Cr3+ ions have unpaired d electrons, which require spin-polarized

DFT calculations. Another important aspect of the calculations is that the CrxAl1−xN alloys are paramagnetic [120, 125, 126], and this state has to be captured explicitly in the DFT calculations. Therefore, in addition to the conﬁgurational disorder simulated via SQS, the paramagnetic state requires truly random conﬁgurations for the spins associated with the Cr3+ ions [127, 128]. However, as shown by Abrikosov et al. [128], the paramagnetic state can be approximated by using disordered, collinear, static spins because such state yields

zero spin-spin correlation functions. To represent the paramagnetic state of CrxAl1−xN, for

n a given alloy structure with n Cr sites, we performed a minimum of 2 and maximum 20 calculations. In these calculations, the spins on Cr sites are randomly initialized subject to the restriction of zero total spin for each concentration and each SQS structure. An example of such a random distribution of initial spins is illustrated in Figure 2.1 for x = 25% Cr concentration.

2.2.2 Details of the DFT Calculations

Structural optimizations and calculations of piezoelectric and elastic constants were carried out using the Vienna Ab-initio Simulation Package (VASP) [129], with projector augmented waves (PAW) in the generalized gradient approximation using the Perdew-Burke- Ernzerhof (PBE) exchange-correlation function [130] and an on-site Hubbard term [131] U for the Cr 3d states. The plane wave cutoﬀ energy was set to 540 eV in all calculations. For the wurtzite structures, we have used 4 4 2 (128 atoms) and 2 2 2 (32 atoms) × × × ×

34 Configuration-1 Configuration-2 Configuration-3

Cr Al

Configuration-4 Configuration-5 Configuration-6

Figure 2.1: Schematics of cation sublattice of CrxAl1−xN alloy. Al (Cr) sites are shown as gray (green) spheres. At a given Cr concentration, the Cr sites of each conﬁguration have a diﬀerent and random spin initialization with zero total spin in order to capture the paramagnetic state.

SQS supercells; for the rocksalt structures, the computations were carried out on 2 2 2 × × (64 atoms) SQS supercells. Brillouin zone sampling was performed by employing 1 1 1 × × and 2 2 2 Monkhorst-Pack [132] k-point meshes for the wurzite and rocksalt structures, × × respectively, with the origin set at the Γ point in each case. Piezoelectric coefficients were calculated using density functional perturbation theory, and the elastic constants were computed by finite differences [133, 134]. The on-site Coulomb interaction for Cr atoms was set at 3 eV, through a Dudarev approach [131]. Before performing the calculations for elastic and piezoelectric constants, we performed cell shape, volume, and ionic relaxations in order to obtain the equilibrium lattice parameters and ionic positions at each particular Cr

35 concentration and SQS alloy.

2.2.3 Experimental Procedures

Combinatorial synthesis of CrxAl1−xN films was performed through reactive physical vapor deposition (PVD). Two inch diameter circular aluminum (99.9999%) and chromium (99.999%) metallic targets were arranged at 45o angles measured from the normal to a plasma-cleaned Si(100) substrate inside a custom vacuum system with a base pressure of 5 10−6 torr. Magnetron RF sputtering with a power of 60 W for aluminum targets and 40 × W for the chromium targets was performed at a deposition pressure of 3 10−3 torr, with 8 × sccm of argon and 4 sccm of nitrogen, and a substrate temperature of 400 oC. Aluminum glow discharges were oriented opposite to each other, with the chromium target perpendicular to both, resulting in a film library with a compositional range in one direction [135, 136]. Each sample library was subdivided into eleven regions across the composition gradient, which were subsequently characterized by x-ray diffraction (XRD) and x-ray fluorescence (XRF), performed on a Bruker D8 Discovery diffractometer with a 2D area detector in a theta-2theta configuration and a Fischer XUV vacuum x-ray spectrometer, respectively.

2.3 Results and Discussion 2.3.1 Enthalpy of Mixing

The enthalpy of mixing as a function of the Cr concentration x, at zero pressure, is deﬁned with respect to the pure wurzite-AlN and rocksalt-CrN phases via

∆Hmix(x)= ECrxAl1−xN xErs-CrN (1 x)Ew-AlN, (2.1) − − − where ECrxAl1−xN, Ers−CrN, and Ew−AlN are the total energies per atom of the SQS alloy, pure AlN phase, and pure CrN phase, respectively . The DFT calculated mixing enthalpies for the wurtzite and rocksalt phases of CrxAl1−xN are shown in Figure 2.2(a). The wurzite phase is found to be favorable up to x = 0.25, beyond which rocksalt alloys are stable; this wurzite to rocksalt phase transition point is consistent with previous experimental observations and other theoretical predictions [120, 121].

36 We have compared the mixing enthalpy of the CrxAl1−xN alloys with that of several

other common wurzite-based nitrides [76, 137, 138, 105, 119], ScxAl1−xN, YxAl1−xN, and

YxIn1−xN, with the results shown in Figure 2.2(b). The mixing enthalpies are positive for all cases, meaning that the alloying of AlN or InN with their respective end members is an endothermic process. In practice, these alloys are formed as disordered solid solutions obtained using physical vapor deposition techniques operating at relatively low substrate temperatures because of the energetic plasmas involved [118, 139]. Figure 2.2(b) shows that the mixing enthalpy in CrxAl1−xN lies between values corresponding to other systems

synthesized experimentally, hence CrxAl1−xN is no more diﬃcult to synthesize than the others. More importantly, the enthalpy calculations show that the transition to rocksalt occurs at the lowest alloy concentration across the wurtzite systems considered, which is important for achieving maximum piezoresponse-enhancing structural frustration with a minimum of dopant concentration in order to retain the single-phase wurtzite.

2.3.2 Piezoelectric Stress Coeﬃcients

The piezoelectric coefficients eij for different spin configurations in SQS supercells with the same Cr content are shown in Figure 2.3. For clarity, the panels in Figure 2.3 are arranged in the same fashion as the piezoelectric tensor when represented as a matrix in

Voigt notation. The vertical scale is the same for all coefficients except e33,e31, and e32. The scatter in the results corresponds to different SQS supercells at each Cr concentration; this is an effect of the finite size of the system, in which local distortions around Cr atoms lead to small variations of the lattice constants and angles. It is for this reason that we average the SQS results at each Cr concentration, thereby obtaining smoother variations of

the piezoelectric coeﬃcients. At 25% Cr, the value of e33 becomes 1.7 times larger than ∼ that corresponding to pure AlN.

37 (a)0.5 Wurzite (B4) Rocksalt (B1) w-CrxAl1-xN 0.4 rs-CrxAl1-xN

0.3 (eV/atom) 0.2 mix H Δ 0.1

0 0 10 20 30 40 50 60 70 80 90 100 (b) Cr (%) 0.6

0.5 w-YxAl1-xN

0.4

w-CrxAl1-xN 0.3 w-Sc Al N (eV/atom) x 1-x

mix 0.2 H Δ w-Y In N 0.1 x 1-x

0 0 10 20 30 40 50 60 70 80 90 100 x (%) Figure 2.2: (a) DFT-calculated mixing enthalpies of the wurtzite and rocksalt phases of CrxAl1−xN as functions of Cr concentration. (b) Calculated mixing enthalpies for several wurzite-based nitride alloys grown experimentally.

random average (vertical axis, C/m2; horizontal axis, % Cr) 1.00 1.00 e 1.00 e 1.00 e 1.00 e 1.00 e e11 12 13 14 15 16 0.75 0.75 0.75 0.75 0.75 0.75

0.50 0.50 0.50 0.50 0.50 0.50

0.25 0.25 0.25 0.25 0.25 0.25

0 0 0 0 0 0

−0.25 −0.25 −0.25 −0.25 −0.25 −0.25

−0.50 −0.50 −0.50 −0.50 −0.50 −0.50

−0.75 −0.75 −0.75 −0.75 −0.75 −0.75

−1.00 −1.00 −1.00 −1.00 −1.00 −1.00 0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25 1.00 1.00 1.00 1.00 1.00 1.00 e21 e22 e23 e e25 e26 0.75 0.75 0.75 0.75 24 0.75 0.75

0.50 0.50 0.50 0.50 0.50 0.50

0.25 0.25 0.25 0.25 0.25 0.25

0 0 0 0 0 0

−0.25 −0.25 −0.25 −0.25 −0.25 −0.25

−0.50 −0.50 −0.50 −0.50 −0.50 −0.50

−0.75 −0.75 −0.75 −0.75 −0.75 −0.75

−1.00 −1.00 −1.00 −1.00 −1.00 −1.00 0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25 0.25 0.25 2.8 1.00 1.00 1.00 e31 e32 e33 e34 e35 e 2.6 0.75 0.75 0.75 36 0 0 2.4 0.50 0.50 0.50

−0.25 −0.25 2.2 0.25 0.25 0.25

−0.50 −0.50 2.0 0 0 0

1.8 −0.25 −0.25 −0.25 −0.75 −0.75 1.6 −0.50 −0.50 −0.50 −1.00 −1.00 1.4 −0.75 −0.75 −0.75

−1.25 −1.25 1.2 −1.00 −1.00 −1.00 0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25 Figure 2.3: The 18 components of the piezoelectric tensor calculated for SQS supercells with different spin configurations and compositions of Cr. At each Cr composition, the open circles represent data for each SQS cell used, while the red solid circles represent the average values across the random initial configurations.

39 random average

(a) 2.8 (b)−0.1 (c) 3.00

2.6 2.75

2.4 −0.2 ) 2 ) 2.50 2

) 2.2 2 (C/m (C/m 2.0 −0.3 2.25 (C/m 33 e

1.8 clamped 33 e

non-clamped 33 2.00 e 1.6 −0.4 1.75 1.4

1.2 −0.5 1.50 0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25 (d)3.50 (e)0.30 (f) 1.62

3.25 0.27 1.60

3.00 0.24 1.58 ε

* 33 2.75 c/a Z du/d 0.21 1.56 2.50

0.18 1.54 2.25

2.00 0.15 1.52 0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25 Cr (%) Cr (%) Cr (%)

total clamped nonclamped Figure 2.4: Variation of (a) e33 , (b) e33 component of e33, (c) e33 , (d) 33 component of the Born eﬀective charge tensor, (e) strain sensitivity of internal parameter, (f) c/a ratio with respect to Cr addition.

40 (a)0.44 (b)

0.42 N uCr

u 0.40 c Al u Al uc 0.38

0.36 5 10 15 20 25 Cr (%)

Figure 2.5: (a) Variation of internal parameter, u with Cr addition. (b) Crystal structure of wurzite AlN.

The piezoelectric coeﬃcient e33 of wurzite CrxAl1−xN is shown Figure 2.4(a) as a function of Cr concentration, and can be written as [101]

clamped non-clamped e33(x)= e33 (x)+ e33 (x), (2.2)

clamped in which e33 (x) describes the electronic response to strain and is evaluated by freezing non-clamped the internal atomic coordinates at their equilibrium positions. The term e33 (x) is due to changes in internal coordinates, and is given by

∗ non-clamped 4eZ33(x) du(x) e33 (x)= (2.3) √3a(x)2 dǫ where e is the (positive) electron charge, a(x) is the equilibrium lattice constant, u(x) is

∗ the internal parameter of the wurtzite, Z33(x) is the dynamical Born charge in units of e, non-clamped and ǫ is the macroscopic applied strain. e33 (x) describes the piezoelectric response coming from the displacements of internal atomic coordinates produced by the macroscopic strain. Based on Eqs. (2.2) and (2.3), panels (b) through (f) in Figure 2.4 show the diﬀerent

relevant quantities contributing to e33 in order to identify the main factors responsible for the increase of piezoelectric response with Cr addition. Direct inspection of Figures Figure 2.4(a- c) indicates that the main contribution to the increase of e33 comes from the non-clamped

41 ∗ ionic part, Figure 2.4(c). Since the Born charge Z33 [Figure 2.4(d)] is practically constant, the key factor that leads to increasing the piezoelectric coeﬃcient is the strain sensitivity du/dǫ of the internal parameter u [Figure 2.4(e)]. Although the internal parameter u is an average value across the entire supercell, the individual average u parameters can also be determined separately for AlN and CrN tetrahedra [Figure 2.5(a,b)]. The internal parameter u of AlN tetrahedra [Figure 2.5(b)] does not change signiﬁcantly, while that of the CrN tetrahedra grows approximately linearly with Cr concentration [Figure 2.5(a)]. In an alloy system where AlN tetrahedra are the majority, this variation can be understood based on (i) the fact that the ionic radius of Cr is about 10% larger than that of Al, and (b) the increase in Cr concentration will lead to average u parameters mimicking the variation of the u parameter corresponding to CrN tetrahedra.

2.3.3 Comparison with Other Wurtzite-Based Alloys

3.0 CrxAl1-xN ScxAl1-xN 2.5 YxAl1-xN YxIn1-xN )

2 2.0 (C/m

33 1.5 e

1.0

0.5 0 5 10 15 20 25 x (%)

Figure 2.6: Comparison of change in e33 with addition of diﬀerent transition metals for x 25% regime. ≤

42 The results from calculations of the piezoelectric properties of CrxAl1−xN with x from

0 to 25% Cr are plotted in Figure 2.6, together with the calculated values for ScxAl1−xN,

2 YxAl1−xN, and YxIn1−xN. In CrxAl1−xN, e33 increases rapidly from 1.46 to 2.40 C/m for Cr concentrations from 0 to 25%. For all other alloys considered, the increase is smaller in the same interval of solute concentration: for ScxAl1−xN, YxAl1−xN, and YxIn1−xN, e33 increases, respectively, from 1.55 to 1.9 C/m2, 1.55 to 1.7 C/m2, and 0.9 to 1.2 C/m2. Within the x 25% range, Cr is more eﬀective than any of the other studied transition elements in ≤ improving piezoelectric response of AlN-based alloys.

(a) 400 CrxAl1-xN ScxAl1-xN 350 YxAl1-xN YxIn1-xN 300

(GPa) 250 33 C 200

150

100 (b) 0 5 10 15 20 25 20 CrxAl1-xN ScxAl1-xN YxAl1-xN 15 YxIn1-xN Exp-CrxAl1-xN

10 (pC/N) 33 d

0 0 5 10 15 20 25 x (%)

Figure 2.7: Variation of (a) C33 and (b) d33 for several nitride-based wurtzite alloys.

The experimentally measurable property is d33, which is commonly known as piezoelectric strain modulus and relates the electric polarization vector with stress. The relationship

43 between the piezoelectric strain and stress moduli is [15]

6 −1 dij = eik(C )kj, (2.4) Xk=1

where Cij are the elements of the stiﬀness tensor in Voigt notation. The variation of the

elastic constant C33 in CrxAl1−xN with x is shown in Figure 2.7(a), along with the other systems considered here. For all of these wurtzite-based piezoelectrics, the increase in piezoelectric response with alloying element concentration is accompanied by mechanical softening

(decrease in C33). From Eq. (2.4), it follows that the increase in e33 [Figure 2.6] and the

mechanical softening [Figure 2.7(a)] cooperate to lead to the increase of d33 values with alloy concentration x. Our calculated d33 values for CrxAl1−xN are in good agreement with experimental data from Ref. [139] [Figure 2.7(b)] for Cr concentrations up to 6.3%. Beyond this concentration, Luo et al. [139] report a drop in the d33 values of their films, which is attributed to changes in film texture. We have also extended our calculations of piezoelectric coefficients beyond 25% Cr composition in wurtzite structures. Figure 2.8 shows that e33 continues to increase at least up to 37.5% Cr. The calculations done at 50% Cr, which start with wurtzite SQS configurations, evolve into rocksalt configurations during relaxation, which explains the decrease of e33 to zero in Figure 2.8. It is worthwhile to compare the performance of several AlN wurzite-based materials for their use in applications. These applications, which are mainly resonators, ultrasound wave generators, GHz telecommunications, FBAR devices, bulk or surface acoustic generators, and biosensors, lead to a multitude of application-specific figures of merit for different utilization modes of the piezoelectric material. However, most figures of merit rely on the piezoelectric properties e33 and d33, both of which in general should be as large as possible for increased piezoelectric device sensitivity. The most used wurtzite material for these applications is AlN, although there are several other options as well (refer to Table I). Alloying with ScN is promising in that it offers an increased d33 for about 10% Sc concentration; larger Sc concentrations are possible, but the growth process becomes more costly and the material

44 xeiet dt onsi iue27) esrmnso ﬁg of Measurements 2.7b). Figure in points (data experiments izeeti rpris eryqarpigtevleof value can the AlN quadrupling of nearly alloying properties, CrN piezoelectric that indicate co results Y Our (14% InN 2.1]. [Table YN-doped and Options content, Y content. 6% Sc at improvement increased marginal with texture lose to likely is 50%. etrta hs fAN especially AlN, of those than properti better piezoelectric the Consequently, for phase. signiﬁcant rocksalt without point non-equilibriu transition the the past Sec.2.3.4, content in Cr see shall we As piezoelec generators. CrN-alloyed new, of conten synthesis alloy for candidate higher prime require Cr (which the materials other makes the above, o with mentioned 2.2] technologies [Figure lowest the the for is relevant point transition the that fact The iue28 aito of Variation 2.8: Figure

2 e33 (C/m ) − 0.5 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0 02 04 50 40 30 20 10 0 e 33 sfnto fC ocnrto o utiepaealy up alloys phase wurtzite for concentration Cr of function as d 33 rfrt al .) ned hsi onotin out born is this Indeed, 2.1). Table to (refer 45 Cr (%) d saeepce ob significantly be to expected are es 33 aino h (non-piezoelectric) the of mation rc o eoaosadacoustic and resonators for trics rwhtcnqe a bring can techniques growth m Tbe21 ihrsett AlN. to respect with 2.1] [Table )adhnernesC-l a Cr-AlN renders hence and t) uha loigwt Noffer YN with alloying as such l utiebsdmaterials wurtzite-based all f avg-32-atom cell 32-atom cell avg-128-atom cell 128-atom cell ec ueirvle o the for values superior reach loigese compared easier alloying N rso ei o specific for merit of ures tn)frssimilarly fares ntent) to device configurations will be needed in the future, as those require not only combinations of elastic and piezoelectric properties, but dielectric properties as well [105].

Table 2.1: Piezoelectric properties of AlN and a few wurtzite alloys for piezoelectric device applications.

2 Material e33 (C/m ) d33 (pC/N) Refs. AlN 1.55 4.5-5.3 [106,107,115] Sc-AlN,10%Sc 1.61 7.8 [140,141] Y-AlN, 6% Y 1.5 4.0 [119] Y-InN,14%Y 1.1 5.1 [137] this work: Cr-AlN,12.5%Cr 1.84 9.86 Cr-AlN,25.0%Cr 2.35 16.45 Cr-AlN,30.0%Cr 2.59 19.52

2.3.4 Experimental Results

To bring experimental support to our proposal that the Cr-AlN system can become a key piezoelectric material to replace AlN and perhaps even ZnO for future applications, we have to ensure that the texture obtained during growth is stable for sufficiently high CrN concentrations. After synthesizing Cr-AlN alloys through reactive PVD, we have performed transmission electron microscopy (TEM) analysis of the films grown in order to check for textural integrity (i.e., grains oriented primarily with the c axis close to the surface normal) and for the onset of the rocksalt phase. At CrN content below 25% [the theoretical boundary shown in Figure 2.2(a)], our films display no significant texture variations. For example, Figure 2.9(a) shows a typical TEM micrograph wherein texture is preserved over the film thickness. Additionally, our energy dispersive spectrocopy (EDS) characterization shows nearly constant Cr content through the sample [Figure 2.9(b)]. Further characterization by XRD was performed for all CrN compositions in the combinatorially synthesized films.

Figure 2.10 shows the XRD results for the 88 discrete CrxAl1−xN compositions produced in an eﬀort to test the possibilities of synthesizing alloys in a wide range of concentrations, including alloys beyond the wurtzite-to-rocksalt transition point. At low alloying levels,

46 (a)

0.10 N

x (b) Cr

1-x 0.08

0.06

(%) in Al in (%) x

Distance (mm)

Figure 2.9: Representative transmission electron microscopy (TEM) image (a) and an energy dispersive spectroscopy (EDS) line scan (b) of an (Al1−xCrx)N ﬁlm cross section containing 7% Cr, conﬁrming the incorporation of Cr into wurtzite solid solution. EDS data were collected∼ along the dashed black line shown in panel (a).

the films grow exclusively with the wurtzite structure and a (002) preferred orientation, as indicated by the dominant presence of the wurtzite (002) diffraction peak [Figure 2.10, left side]. Films grown by reaction PVD under the conditions used here accept chromium into the wurtzite lattice and grow primarily with the ideal (002) orientation. With increased CrN content, the wurtzite (012) and (010) peaks appear, indicating some deviations from the original, and still predominant (002) orientation of the film. The metastability of this alloy is overcome at an approximate composition of x 30%, where the polycrystalline ≃ rocksalt phase appears, as revealed by the rocksalt (002) and (111) peaks (Figure 2.10, right hand side). These experimental results show that wurtzite CrxAl1−xN solid solutions can be synthesized without observable phase separation up to concentrations of 30% Cr. Wurtzite material still exists at global compositions beyond 30%, but in a wurtzite-rocksalt phase mixture, which will diminish the piezoelectric properties because of the presence of a significant amount of centrosymmetric rocksalt phase in the mixture.

47 Figure 2.10: X-ray diffraction patterns of the thin film combinatorial libraries plotted against the film composition, with comparison to the patterns for wurtzite (WZ) [142] and rocksalt(RS) [143] structures. For alloying content x< 30%, the films grow predominantly with the wurtzite structure. At higher Cr concentrations, x > 30%, both rocksalt and wurtzite phases are detected, and the wurtzite exhibits degraded texture. No films were produced with compositions in regions where no intensity is shown.

There are few studies of Cr alloyed into wurtzite AlN, [139, 144, 126] reporting Cr-doped alloys grown by magnetron sputtering. The Cr concentration previously attained is below 10%, although the limits of Cr alloying were not actually tested in the previous reports [139, 144, 126]. Our combinatorial synthesis results show that Cr can be doped into the wurtzite lattice up to 11% before the predominant (002) ﬁlm texture starts to change, and up to 30% before the rocksalt phase appears.

2.4 Concluding Remarks

By using a physical representation of the paramagnetic state of substitutional Cr in a wurtzite AlN matrix and performing the necessary averaging over spin conﬁgurations at each Cr concentration, we computed the structural, mechanical, and piezoelectric properties of (Cr, Al)N alloys. Our combinatorial synthesis experiments showed that (Cr, Al)N are relatively easy to synthesize, and also showed that the reactive PVD procedure resulted in (Cr, Al)N alloys retaining the wurtzite structure for alloying concentrations up to 30% Cr.

Remarkably, our DFT calculations of piezoelectric properties revealed that for 12.5% Cr d33

48 is twice that of pure AlN, and for 30% Cr this modulus is about four times larger than that of AlN. From a technological standpoint, this finding should make (Cr, Al)N the prime candidate to replace the current-wurtzite based materials in resonators and acoustic wave generators. The larger piezoelectric response (than AlN) may lead to smaller power consumption and perhaps even to avenues to further miniaturize various devices. While the substitutional alloying with Cr would improve the piezoelectric response for every type of device in which currently AlN is being used, one may wonder why not alloying with other trivalent metals, such as Y or Sc. In particular, Sc has been shown to significantly increase the piezoelectric modulus as well [102]. Even though (Sc, Al)N has more exciting properties [103, 102] than (Cr, Al)N, the reason why (Sc, Al)N alloys have not taken over the resonator market so far is that the outstanding enhancements in piezoelectric properties occur at very high Sc concentration (Figure 2.2, x > 55%), at which the stability of the wurtzite phase is rather poor. (Cr, Al)N has a low wurtzite-to-rocksalt transition concentration, and therefore can offer certain piezoelectric enhancements at alloying levels that are easier to stabilize during the synthesis. In order to ensure significant impact of (Cr, Al)N alloys as materials to outperform and replace the established piezoelectrics AlN and ZnO, two avenues should be pursued in the near future. First, to benefit from the 300% increase in d33 at 30% Cr content, it is not sufficient that the rocksalt phase does not form up to that Cr concentration: we also have to avoid the formation of (012) and (010)-oriented grains during growth, which would downgrade (simply through directional averaging) the piezoelectric enhancements associated with the (002)-oriented grains. To that end, we envision changing substrates so as to enable better lattice matching with Cr-alloys with over 25% Cr. This can effectively prevent the (012) and (010) textures from emerging, therefore creating the conditions to take advantage of the large increase in d33 reported here. Second, future experimental efforts should measure device performance especially to understand the additional aspect of how Cr content in

49 wurtzite affects the bandgap and whether there would be deleterious leakage effects at larger Cr concentrations. Assuming a worst case scenario, these effects can be mitigated by co- alloying with a non-metallic atomic species (e.g. boron). Pursuing the two directions above can make (Cr, Al)N suitable for simultaneous optical and mechanical resonators, [145, 146] which are relatively new applications that currently exploit multi-physics aspects of AlN. At present, the characterization of (Cr, Al)N for these multifunctional applications that require simultaneous engineering of the photonic and acoustic band structure is rather incipient, and only few relevant properties of the (Cr, Al)N alloys are known: for example, for a Cr concentration of about 2%, the bandgap is virtually un- changed, while the adsorption band decreases from 6 to 3.5 eV [?]. Future theoretical and experimental work to investigate, e.g., photoelastic effect and optical attenuation, is necessary in order to fully uncover the potential of Cr-doped AlN for these applications. For now, we surmise that the technological reason for which one would replace AlN with (Cr, Al)N for use in multifunctional resonators is the trade-off between the increase in vibrational amplitude and decrease in frequency: while low amounts of Cr may lower the frequency somewhat, the oscillation amplitude would increase due to larger piezoelectric response. The decrease in frequency can be mitigated by co-doping with a small trivalent element (boron), as shown for other doped AlN alloys [105]. Last but not least, it is worth noting that doping with Cr could enable magnetic polarizaton of the Cr ions in the wurtzite lattice and/or of the minor- ity carriers:these effects are non-existent in pure AlN, and could be pursued for spintronic applications or for low-hysteresis magnets [126]. The significant increase of piezoelectric modulus reported here provides significant drive to pursue the two directions identified above, and overcome routine barriers towards estab- lishing (Cr, Al)N as a replacement for AlN with large performance enhancements. Acknowledgments. The authors gratefully acknowledge the support of the National Sci- ence Foundation through Grant No. DMREF-1534503. The DFT calculations were performed using the high-performance computing facilities at Colorado School of Mines (Golden

50 Energy Computing Organization) and at National Renewable Energy Laboratory (NREL). Synthesis and characterization facilities at NREL were supported by the US Department of Energy, Oﬃce of Science, Oﬃce of Basic Energy Sciences, as part of the Energy Fron- tier Research Center “Center for Next Generation of Materials by Design: Incorporating Metastability” under contract No. DE-AC36-08GO28308.

51 CHAPTER 3 TUNING THE PIEZOELECTRIC AND MECHANICAL PROPERTIES OF THE AlN SYSTEM VIA ALLOYING WITH YN AND BN

A paper published in Journal of Applied Physics5 Sukriti Manna6, Geoff L Brennecka7, Vladan Stevanović7, Cristian V Ciobanu6 Recent advances in microelectromechanical systems often require multifunctional materials, which are designed so as to optimize more than one property. Using density functional theory calculations for alloyed nitride systems, we illustrate how co-alloying a piezoelectric material (AlN) with different nitrides helps tune both its piezoelectric and mechanical properties simultaneously. Wurtzite AlN-YN alloys display increased piezoelectric response with YN concentration, accompanied by mechanical softening along the crystallographic c direction. Both effects increase the electromechanical coupling coefficients relevant for transducers and actuators. Resonator applications, however, require superior stiffness, thus leading to the need to decouple the increased piezoelectric response from the softened lattice. We show that co-alloying of AlN with YN and BN results in improved elastic properties while retaining some of the piezoelectric enhancements from YN alloying. This finding may lead to new avenues for tuning the design properties of piezoelectrics through composition-property maps.

3.1 Introduction

Recent technological advances in microelectromechanical systems require materials to be more efficiently designed, often leading to optimization of more than one property [147, 148]. An electromechanical (piezoelectric) material requires the best piezoelectric coefficient, while materials for damping high mechanical loads require the high stiffness moduli [149, 14].

5Reprint of Journal of Applied Physics 122 (10), 105101 (2017) with permission from American Institute of Physics 6Department of Mechanical Engineering, Colorado School of Mines, Golden, CO 80401, USA 7Department of Metallurgical and Materials Engineering, Colorado School of Mines, Golden, CO 80401, USA

52 The high stiffness value is also associated with thermal stability, enabling the use of these materials over a large temperature range [71]. Pseudo-binary alloys, i.e., solid solutions between two compounds, have been extensively investigated for use in electromechanical applications [102, 150, 103, 119, 137]. Most of these alloys have large miscibility gaps, which makes them less appealing for applications where mechanical stability is an important figure of merit. The presence of miscibility gaps is not particularly restrictive, since current non- equilibrium processing techniques allow for fabricating pseudo-binary alloys in which one can seek a given functionality by, e.g., changing composition (often through the miscibility gaps) so as to optimize piezoelectric coefficients, stiffness, electromechanical coupling constants, or electronic properties. However, achieving optimal or desired values for more than one such property is challenging since optimizing one property often changes another towards undesirable values. In such cases, more degrees of freedom may help create a framework in which advanced materials can be optimized for multiple functionalities. In this article, we focus on a piezoelectric system to show that alloying with more than one compound enables us to tune two properties, specifically, the piezoelectric coefficient and the mechanical stiffness. With the discovery of the anomalously large increase in the piezoelectric moduli of wurtzite AlN when alloyed with ScN [76] or YN [119, 151], AlN-based alloys have received a lot of attention as materials for improved telecommunications, sensors, and surface acoustic wave devices [32, 152]. The origin of the enhanced piezoelectricity was revealed to be an intrinsic effect of alloying, as opposed to textural or microstructural effects [102], with the obvious implication that controlling the piezoelectric enhancement can be done mainly by increasing composition of YN or ScN. However, such alloying softens the material significantly [102], which actually makes the material less attractive for resonant applications for which k2Q is a common figure of merit [153] wherein k is an electromechanical coupling coefficient and Q is a quality factor that is inversely proportional to mechanical dissipation [153]. It is therefore desirable to control the mechanical properties as well, which could in principle be achieved by alloying with other species.

53 Herein, we focus on AlN-YN alloys and advance the premise that further alloying this system with BN can increase the elastic constants of the material, thus leading to new avenues for piezoelectric design based on precise control YN and BN compositions. The choice for YN is motivated by the scarcity of literature reports on AlN-YN alloys [151, 154], and by the fact that Y may be a less expensive alternative for Sc. While the elastic stiffening of AlN-YN upon alloying with BN would be enabled by the presence of shorter and stiffer B-N bonds, the effect of alloying with BN on the piezoelectric properties is presently unknown. Using density functional theory (DFT), we have found that the presence of BN, by itself, indeed

leads to increases in the relevant elastic constant C33, but does not signiﬁcantly change the

piezoelectric coefficient e33. Furthermore, we have uncovered the combined effect of alloying with YN and BN on the properties of the AlN system for total alloying concentrations up to 50 at.%. The addition of BN to AlN-YN alloys counters to some extent the piezoelectric enhancement obtained via YN alloying, but increases the elastic modulus. Overall, the alloying of AlN with BN and YN leads to a high degree of control over the piezoelectric and elastic properties of the resulting alloy, and therefore over the coupling coefficients and various figures of merit commonly used in piezoelectric device design.

Figure 3.1: Atomic structure of a representative SQS Al50Y25B25N alloy structure

54 3.2 Methods

To assess the changes in elastic and piezoelectric properties driven by diﬀerent levels of

YN and BN in Al1xyYxByN alloys, we ﬁrst performed DFT relaxations of atomic positions and lattice constants using the VASP package [129], with projector augmented waves [130] in the generalized gradient approximation using the Perdew-Burke-Ernzerhof (PBE) exchange- correlation functional [155]. Out of caution more than necessity, we introduced an on-site Coulomb interaction (parameter U = 3 eV) for the yttrium d state since this state is explicit in the pseudopotential. When compared with calculations done with plain PBE (U = 0 eV), the results for lattice constants are nearly identical, while the piezo and stiﬀness tensor component vary by less than 2%.

Figure 3.2: [(a) and (b)] Lattice constants a and c for AlN-YN and AlN-BN alloys as a function of dopant (Y or B) cation concentration x. Panel (c) shows the c/a ratio for these alloys.

The parent structure is wurtzite AlN, in which Al is substituted with Y and B in desired amounts. Supercells with 32 atoms are simulated using special quasirandom structures [SQS, more details in Appendix D] [124] ( Figure 3.1), whose pair correlation functions are the same as those of random alloys up to second-nearest neighbors. A 5 5 3 Monkhorst- × × Pack [132] k-point mesh was used to sample the Brillouin zone, resulting in 38 irreducible k-points. A plane wave cutoﬀ energy of 500 eV was used; the energy tolerance stopping criteria were 10−8 and 10−7 for the electronic and ionic convergence, respectively. These

55 criteria were obtained by progressive tightening of the convergence until the relaxed SQS supercells stopped changing. This resulted in smoother curves for lattice parameters and properties, in which the only source of noise remained the use of small number of atoms coupled with a limited number of individual SQS supercells at each concentration (refer to Fig. SM-1 FI for an assessment of the dispersion of individual SQS values). Due to the variability of atomic-scale environments in the SQS supercells [103], we performed ﬁve calculations at each composition and used them to average the properties, including the lattice constants. For the separate alloying only with YN or with BN, the lattice constants a and c are plotted in Figure 3.2(a) and Figure 3.2(b) as functions of the cation concentration. Alloying with YN increases the lattice constants, while alloying with BN alone decreases them: these trends are simply due to the larger (smaller) cation radius of Y (B) compared to that of Al in AlN. The c/a ratio is more sensitive to doping for the case of Y compared to BN [Figure 3.2(c)]. Our DFT calculations of mixing enthalpies for alloy wurtzite and rocksalt phases of AlN- YN revealed that wurtzite is more stable than rocksalt for up to 72% Y content, above which rocksalt becomes more stable [Figure 3.3(a)]. The mixing enthalpies in Figure 3.3(a) are positive, meaning that mixing is not spontaneous. From the mixing enthalpies and combinatorial expression for mixing entropy in this pseudo-binary alloy, we have obtained the temperature composition phase diagram in a manner similar to that described by, e.g., Burton et. al. [156] for other wurtzite systems. This phase diagram is shown in Figure 3.3(b), in which the temperature axis is rather qualitative since we have not considered lattice vibrations in deriving it; it is similar to the one that would be obtained in the presence of vibrations [156]. As would be expected, the diagram shows a miscibility gap-the white space between the metastable regions in Figure 3.3(b). This means that compositions in that gap cannot be achieved under thermodynamic equilibrium. However, the current growth techniques (e.g., reactive dc magnetron co-sputtering) [157, 104] rely on non-equilibrium processes to overcome the solubility limits [158, 159, 160] and create alloys with better control over properties

56 (a) 1.0 w-YxAl1-xN 0.8 rs-YxAl1-xN w rs

0.6 (eV/f.u) 0.4 mix H Δ 0.2

0 0 20 40 60 80 100 (b) 4500 w rs 3750 metastable

3000

2250 metastable

1500

Temperature (K) Temperature 750

0 0 20 40 60 80 100 Y (%)

Figure 3.3: Calculated mixing enthalpy curves for AlN alloyed with YN in the wurtzite (w) and rocksalt (rs) structures. through control of composition.

3.3 Results and Discussions

Focusing now on describing the piezoelectric properties of the alloys, we note that the piezoelectric coeﬃcient e33 is evaluated at each Y(x) and B(y) composition via

∗ clamped−ion 4eZ du e33 = e33 + 2 (3.1) √3a0 dǫ clamped−ion ∗ where, e33 is the clamped-ion contribution to e33, e is the electron charge, Z is the axial component of the dynamical Born effective charge tensor, a0 is the equilibrium lattice constant, and du/dǫ is the sensitivity of the internal parameter u to strains along the crystallographic c-axis [161]. The elastic constants were computed by finite differences

57 between the energies of the lattices subjected to small applied strains [162].

(a) 3.5 400

350 3.0

300 C ) 2

2.5 33

250 (GPa) (C/m

33 2.0 e 200

1.5 150

1.0 100 0 10 20 30 40 50 Y (%) (b) 3.5 500

3.0 450 C ) 2

2.5 33

400 (GPa) (C/m

33 2.0 e

350 1.5

1.0 300 0 10 20 30 40 50 B (%)

Figure 3.4: The design premise behind co-alloying AlN with YN and BN. (a) Higher YN content in Al1−xYxN increases the piezoelectric coefficient e33 but decreases the elastic modulus C33. (b) Increasing concentration of BN in Al1−yByN has a small effect on the piezoelectric coefficient, but increases the elastic modulus. Thus, co-alloying with YN and BN could lead to increases in both piezoelectric coefficient and stiffness relative to pure AlN.

Figure 3.4 shows the eﬀects of separate BN and YN alloying on the properties of AlN. As seen in Figure 3.4(a) the piezoelectric coeﬃcient e33 increases with Y concentration up to

50%, while the elastic constant C33 decreases; this is consistent with previous works on the similar AlN-ScN system [102, 103]. The physical origin of this eﬀect arises from the larger radius of the Y ion, which changes the internal parameter u of the alloyed wurtzite structure and its sensitivity to strain, du/dǫ [refer to Eq. (3.3)]. The value of e33 that can be achieved

58 by alloying with 50% Y is doubled compared to that of pure AlN (Figure 3.4). Figure 3.4(b) shows that alloying with BN alone increases the elastic modulus, which is consistent to another report [163]. Interestingly, we also note that substituting B for Al does not alter the piezoelectric coefficient significantly up to concentrations of 50% B. When AlN-YN films are deposited, there is usually strain in the lattice stemming from the mismatch with the substrate, changing film texture with alloy concentration, or from insufficient annealing. It is therefore relevant to assess what the effect of this strain is on the piezoelectric coefficient, so we have redone the DFT calculations for the situations were strain is applied in plane (along the a and b directions) or out of plane (along the c direction).

4.0 Compression Tension

3.5 50.0 % Y

3.0 37.5 % Y ) 2 2.5 25 % Y (C/m 33

e 2.0 12.5 % Y

1.5 0.0 % Y

1.0

−0.02 −0.01 0 0.01 0.02 εc 4.0 Compression Tension

3.5

50.0 % Y 3.0 ) 2 2.5 37.5 % Y (C/m

33 25 % Y e 2.0 12.5 % Y 1.5 0.0 % Y 1.0

−0.02 −0.01 0 0.01 0.02 εa

Figure 3.5: Variation of piezoelectric coeﬃcient in Al1−xYxN with strained applied along the a and c axes.

59 The results of these calculations are displayed in Figure 3.5: e33 can increase up to about 3.75 C/m2 (i.e., 2.5 times that of pure AlN) either by alloying with 50% Y and applying 2% tension along the a axis [Figure 3.5(a)], or by alloying with 50% Y and applying 2% compression along the c axis [Figure 3.5(b)]. The opposite trends of e33 with strain along different directions (Figure 3.5) suggest that, for homogeneous and isotropic strain in the AlN-YN lattice, the effects of strain along c and a directions may cancel each other at least partially and thus not affect the piezoelectric coefficient significantly. This is important when alloying with a secondary species, such as BN, which would amount to roughly isotropic strain when B is (nearly) homogeneously doped throughout the AlN-YN lattice. Based on these results, we have set out to analyze how co-alloying with both YN and BN changes properties, aiming to find composition regimes where the piezoelectric coefficient increases while the elastic modulus does not decrease to the same extent as in the case of alloying with Y alone.

Figure 3.6 shows the variation of e33 and C33 with the Y concentration x, at a fixed level of B doping, y (such that x + y 50%). The data points in Figure 3.6 represent averages ≤ over SQS configurations. Other properties (i.e., e31 and C11) are shown as functions of x and y in Fig. SM-2. Our starting premise that the presence of boron can mitigate or control the decrease in elastic modulus C33 brought about by Y alloying-while still retaining some increase of the piezoelectric coefficient is borne out by the results in Figure 3.6. For example, going from 0% B in Al50Y50N to 18.75% B concentration in Al50Y31.25B18.75N results in a 50

MPa increase of the elastic constant C33 with 33% decrease in the piezoelectric coefficient. For ease of reference, some of the results shown in Figure 3.6 are also tabulated in Table 3.1. BN alloying has a strong effect on mechanical properties (Table 3.1): increasing BN content from 31.25% to 50% at the expense of YN concentration nearly doubles the elastic modulus (from 238 GPa to 470 GPa). Next, we discuss the electromechanical coupling coefficients resulting from co-alloying of AlN with YN and BN. These coefficients are key parameters in designing transducers for

60 ceeoees n yocpscmol eur h piezo the require commonly gyroscopes Appli and accelerometers, 71]. [167, applications sensing and harvesting energy conce B 43.75% Y (h) the to of B functions 0.00% (a) as from nitrides ranging Al-Y-B levels of properties axis) iue36 h e izeeti ( piezoelectric key The 3.6: Figure

2 2 2 2 e33 (C/m ) e33 (C/m ) e33 (C/m ) e33 (C/m ) 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 02 04 50 40 30 20 10 0 50 40 30 20 10 0 50 40 30 20 10 0 02 04 50 40 30 20 10 0 (g) 37.50%B (e) 25.00%B (c) 12.50%B (a) 0.00%B Y (%) Y (%) Y (%) Y (%) e 33 e,lf etclai)adeatc( elastic and axis) vertical left red, ,

100 200 300 300 400 400 500 500 100 200 300 300 400 400 500 500 100 200 300 300 400 400 500 500 100 200 300 300 400 400 500 500

33 33 33 33

(GPa) (GPa) (GPa) (GPa) C C C C 61

2 2 2 2 e33 (C/m ) e33 (C/m ) e33 (C/m ) e33 (C/m ) 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 02 04 50 40 30 20 10 0 02 04 50 40 30 20 10 0 50 40 30 20 10 0 50 40 30 20 10 0 lcrcmdu ooeaei the in operate to medium electric ain uha rsuesensors, pressure as such cations (d) 18.75%B (b) 6.25%B (h) 43.75%B 31.25%B (f) Y (%) Y (%) Y (%) Y (%) taina osatboron constant at ntration C 33 re,right green, ,

100 200 300 300 400 400 500 500 100 200 300 300 400 400 500 500 100 200 300 300 400 400 500 500 100 200 300 300 400 400 500 500

33 33 33 33

(GPa) (GPa) (GPa) (GPa) C C C C Table 3.1: Comparison between the properties of some of the Al-Y-B nitrides computed in this work and those of three commercial piezoelectrics.

2 2 Material e33 (C/m ) C33 (GPa) d33 (pC/N) k33 PZT-7ARef.[164] 9.5 131 153 0.67 PZT-4Ref.[164] 13.84 113 225 0.35 K0.5Na0.5NbO3 Ref.[165,166] 4.14 104 80,130 0.51 this work: AlN 1.46 353 5.16 0.12 Al50Y50N 3.02 158.07 37.67 0.57 Al50Y43.75B6.25N 2.51 161.36 22.48 0.47 Al50Y37.50B12.50N 1.7 180.23 12.92 0.27 Al50Y31.25B18.75N 1.85 209.3 13.34 0.27 Al50Y25B25N 1.58 223.4 11.5 0.2 Al50Y18.75B31.25N 1.93 238 13.03 0.26 Al50Y12.5B37.50N 1.54 253.4 8.8 0.17 Al50Y6.25B43.75N 1.73 328 7.7 0.17 Al50B50N 1.34 469.7 3.42 0.08

longitudinal length mode [168], for which the relevant coupling coeﬃcient is

2 2 e33 k33 = 2 (3.2) ǫ33C33 + e33

where e33 is the 33 component of the dielectric tensor. The static dielectric constants vary by about 10%-15% from their value at x =12.5% (separate calculation) for the entire range

of equimolar Y-B doping (i.e., x =25% in Al1−2xYxBxN), so a constant value of ǫ33 =11.0

2 was assumed in computing the coupling coeﬃcients. High k33 coupling factor corresponds to transducers with better axial resolution, broader bandwidth, and higher sensitivity [169]. Actuators based on cantilevers operate in the transverse length mode (bending) [167, 6], for which the relevant coupling coeﬃcient is

2 2 e31 k31 = 2 (3.3) ǫ33C11 + e31 where e31 and C11 are components of the piezoelectric and elastic tensor, respectively. The two coupling coeﬃcients are plotted in Figure 3.7(a) and Figure 3.7(b), for the range of Y and B concentrations considered. High coupling constants require large piezoelectric co-

62 2 (a) k 33 50 0.40

40 0.30 30

0.20 Y (%) Y 20

10 0.10

0 0 10 20 30 40 50 2 (b) k 31 50 0.08

0.04 30

Y (%) Y 20 0.02

10 0.008 0 0 10 20 30 40 50 B (%)

2 2 Figure 3.7: The coupling constants (a) k33 and (b) k31 as functions of the Y and B concentrations in Al1−x−yYxByN with at least 50% Al. efficients and low elastic moduli. Increasing the amount of Y in the system increases the piezoelectric and coupling coefficients but softens the system elastically [102], which is true for both coupling modes addressed here. The two coupling coefficients exhibit similar trends

63 (Figure 3.7), with larger values achieved for higher Y concentrations. However, variations of

2 k31 are signiﬁcantly smaller, and larger values can also be achieved using equiatomic Y:B alloys. With the piezoelectric and elastic tensor components obtained from DFT calculations, it is useful to compare some of the properties of the co-alloyed AlN system with those of known piezoelectrics. Table 3.1, which also includes the longitudinal piezoelectric modulus d33, shows the properties of three well-known piezoelectrics in comparison with a few selected alloy compositions computed here (for completeness, piezoelectric moduli are also plotted in Fig. SM-3 for the entire range of compositions used). Direct inspection shows that AlN co-alloyed with Y and B can achieve parameters comparable to those of commercial piezoelectrics. More importantly, the simultaneous enhancement of e33 and C33 in comparison to the pure AlN values can be seen for boron levels of up 25%, illustrating that co-alloying with diﬀerent species represents a valuable route for simultaneously engineering the piezoelectric and the elastic properties of a parent wurtzite structure such as that of AlN.

3.4 Conclusion

In conclusion, we have shown that co-alloying of AlN with YN and BN leads to superior control of both piezoelectric and mechanical properties. The idea of introducing BN into the systems arose as a way to stiffen the material, which otherwise would have becomes increasingly softer with increased YN content. In the process of assessing the influence of BN and YN composition on properties, we have separately calculated the effect of straining the lattice in different directions, and found that enhancements of the piezoelectric coefficient can be made by applied strain as well: this strain effect could become important for certain substrates, or for electromechanical applications where strain develops in the structure during operation. The results presented here could carry over to other parent phases and alloying phases as well, provided that each alloying agent offers a clear advantage in terms of engineering one specific property. In our case, YN leads to increasing the piezoelectric coefficient and BN leads to increasing the stiffness. The use of both YN and BN offers intrinsic and simultaneous

64 control over the mechanical and piezoelectric properties of the Al-Y-B-N system, even though the maximal value for each individual parameter can be achieved in the presence of only one dopant.

65 CHAPTER 4 LARGE PIEZOELECTRIC RESPONSE OF VAN DER WAALS LAYERED SOLIDS

A paper submitted to Journal of Materials Chemistry C Sukriti Manna8, Prashun Gorai9, Geoff L Brennecka9, Cristian V Ciobanu8, Vladan Ste- vanović9 The bulk piezoelectric response, as measured by the piezoelectric modulus tensor (d), is determined by a combination of charge redistribution due to strain and the amount of strain produced by the application of stress (stiffness). Motivated by the notion that less stiff materials could exhibit large piezoelectric responses, herein we investigate the piezoelectric modulus of van der Waals-bonded quasi-2D ionic compounds using first-principles calculations. From a pool of 869 known binary and ternary quasi-2D materials, we have identified 135 non- centrosymmetric crystals of which 48 systems are found to have d components larger than the longitudinal piezoelectric modulus of AlN (a common piezoelectric for resonators), and three systems with the response greater than that of PbTiO3, which is among the materials with largest known piezoelectric modulus. None of the identified materials have previously been considered for piezoelectric applications. Furthermore, we find that large d components always couple to the deformations (shearing or axial) of van der Waals “gaps” between the layers and are indeed enabled by the weak intra-layer interactions.

4.1 Introduction

Coupling between the mechanical degrees of freedom and the electric polarization in a solid, a hallmark of piezoelectric materials, has found use in applications that range from sensors, resonators, motors, and actuators, to high-resolution ultrasound devices and miniature ﬁlters for cellular communications [81, 170, 171]. Interestingly, only about 10 piezoelectric

8Department of Mechanical Engineering, Colorado School of Mines, Golden, CO 80401, USA 9Department of Metallurgical and Materials Engineering, Colorado School of Mines, Golden, CO 80401, USA

66 materials, including SiO2 (quartz), LiTaO3, LiNbO3, PZT (lead zirconate titanate)-based,

BaTiO3-based, (K,Na)NbO3-based, Bi4Ti3O12-based, AlN, and ZnO, are technologically relevant and cover virtually all of these applications [172, 173, 6]. Expanding the pool of known piezoelectric materials would help broaden the range of applications and allow earth- abundant and non-toxic [174] replacements for materials that are presently in use. Further- more, it would offer more cost- and performance-effective choices beyond a relatively limited set of materials that are at present used for their piezoelectric properties. The need for new piezoelectrics was recognized before and was the main motivation behind recent computational efforts in high-throughput screening of inorganic materials for piezoelectric performance. These include the work on perovskite alloys, [175, 176] and creation of a database of piezoelectric properties of compounds [177]. Materials with reduced dimensionality, such as 2D mono- and multi-layers have also been investigated recently [178, 179, 180, 181, 182, 183]. In virtually all of these works, the quantity of interest was the piezoelectric coefficient tensor eij, which relates the polarization Pi to strain ǫj,[15, 116] (in

Voigt notation, [18] Pi = j eijǫj) and is typically obtained from first-principles calculations. P In this work, we also apply first-principles calculations to screen for candidate piezoelectrics, but instead of the piezoelectric coefficient tensor we consider the piezoelectric mod-

ulus tensor dij, which relates the induced polarization to the applied stress, Pi = j dijσj P [15, 116]. The advantage of using dij is in that it represents an important ﬁgure of merit in a wide range of technological applications [81, 170] and is a more commonly measured piezoelectric property, especially for materials in the thin ﬁlm form. Because the induced polarization depends on the applied stress through a combination of the charge redistribution due to strain and the amount of strain that is produced by the applied stress, the

piezoelectric modulus tensor dij depends on both the piezoelectric coeﬃcient tensor eij and

the elastic tensor Cij as:

6 −1 dij = eik(C )kj. (4.1) Xk=1

67 aeil ihsimilar with materials h usin eaeadesn r:()wehrmtrasth ( materials response whether piezoelectric (a) strong are: exhibit addressing inter are (vdW) we Waals s der questions van low The weak relatively of presence have the to to due expected direction be can which solids, bonded corr the in responses piezoelectric larger exhibit overall aentbe rvosyadesdi ytmtcway. systematic a in addressed previously To been response. not strong have the enabling in interactions vdW of role nmtraswt ag large large with materials in aeil a xii ag izeeti epne smea as response, piezoelectric large exhibit can materials aesadtevndrWassaigbtenthem. between spacing Waals der van the and layers iue41 lutaino us-Dmtra (TaSe material quasi-2D a of Illustration 4.1: Figure o hsrao,w ocnrt u netgto nlayere on investigation our concentrate we reason, this For rmeuto 41,w a ne htlrepeolcrcm piezoelectric large that infer can we (4.1), equation From

“soft” out-of-plane direction e ij hsi lutae nFgr .,weew oiels tffm stiff less notice we where 4.2, Figure in illustrated is This . e ij u loi ytm ihlwstiffness low with systems in also but , d ij Ta a efud n b fti stu,wa sthe is what true, is this if (b) and found, be can ) 68 Se “stiff” in-planedirection “stiff” 2 ,soigteaoi tutr fthe of structure atomic the showing ), sodn direction. esponding ue by sured u nweg,teequestions these knowledge, our qai2)vndrWaals der van (quasi-2D) d tbln oti ls and class this to belong at odulus ins nteout-of-plane the in tiffness cin ewe h layers. the between actions d ij C oprdt stiffer to compared , d ij ij otr(esstiff) (less Softer . a eexpected be can aterials iue42 orlto ewe xeietlymaue lo measured ( experimentally between Correlation 4.2: Figure efidmr hnoetid(8cmons xii piezoelect ( exhibit AlN compounds) of (48 third one than non-centrosymmet more find 135 we identified have we systems, VdW layered large relatively with materials 17 [81, Refs. from sourced is Data ferroele samples. of polycrystalline case in samples single-domain single-crystal, ag epne ti motn ont htnn fteevdW applications. these piezoelectric of none for that previously note to important is It response. large nadto,w on he aee ytm ihpiezoelect PbTiO with of systems layered three found we addition, In opnns efidta nalo hs ytm,lrepiezoe large systems, these of all in that find we components, d 33 u eut ofimteepcain.Tesac a reveale has search The expectations. the confirm results Our fe efrigatoog nlsso h opigbetween coupling the of analysis thorough a performing After n hi lsi ouu ( modulus elastic their and ) d 3 33 ( 55p/) omnyue izeeti aeili resona in material piezoelectric used commonly a pC/N), =5.5 d 33

d (pC/N) 1000 19p/ ,aohretbihdpeolcrcmtra kn material piezoelectric established another ), pC/N =119 33 100 10 1 0 0 0 0 500 400 300 200 100 0 LiIO d C ij PZT 33 LiGaO KN-Mn0.1 BSPT66 opnns u f89cniee iayadternary and binary considered 869 of Out components. 3 .Gencrlsdnt aafo ihrsnl rsasor crystals single either from data denote circles Green ). PbTiO ZnO 3 BaTiO C 69 33 3 (GPa) LiTaO 3 LiNbO ti aeil.Oag ice denote circles Orange materials. ctric 3 GaN 3 ,14 8,16 8,18 189]. 188, 187, 186, 185, 184, 0, eti epnei lascoupled always is response lectric i ouiee agrta that than larger even moduli ric gtdnlpeolcrcmodulus piezoelectric ngitudinal ytm aebe considered been have systems i ouigetrta that than greater moduli ric i rsas u fthose of Out crystals. ric ubro quasi-2D of number a d AlN aiu tesand stress various o applications. tor w o t very its for own d ij to the stress components that imply deformations (axial or shear) of the van der Waals “gaps” between the layers. This is consistent with the fact that the softest elastic constants are related to these deformations. Ultimately, our results point to the layered vdW systems as a rich chemical space for ﬁnding new piezoelectric materials, and introduce elastic properties as additional design criteria for ﬁnding materials with large piezoelectric modulus.

4.2 Computational Methodology

In this section, we discuss the details of the computational methodology used in our calculations, broadly divided into three main subsections. The ﬁrst subsection describes the procedure for identifying quasi-2D structures from the Inorganic Crystal Structure Database (ICSD) [190, 191]. Next, we describe the drawbacks of GGA exchange correlation functionals for predicting the properties of layered materials and our approach to overcome this issue. Then, we provide detailed descriptions of our workﬂow for evaluating the piezoelectric modulus tensors (d) of quasi-2D solids.

4.2.1 Automated Identiﬁcation of Quasi-2D Materials

An essential component of this work is the identification of layered (quasi-2D) vdW materials from ICSD. To accomplish this, we extend the applications of our previously- developed procedure [192] for automated identification of the binary quasi-2D materials from the ICSD to include ternary chemistries. Similar algorithms have been developed and used for the purpose of broad identification of vdW bonded layered systems by others, including the work of Ashton et al., [193] Mounet et al., [194] and Cheon et al. [195] Our procedure relies on a slab cutting routine and bond counting. In the first step, we cut out stoichiometric slabs of a certain thickness for all symmetry inequivalent sets of Miller indices (hkl) within a certain range ( 3 h,k,l 3). Next, for each slab we find − ≤ ≤ the terminations of its surfaces that minimize the number of broken bonds by translating surface atoms from one side of the slab to the other using appropriate lattice vectors. We then count the (minimal) number of broken bonds, i.e., the under-coordination, of the surface

70 atoms. The condition of quasi low-dimensional crystals then implies the existence of (hkl) directions for which the corresponding slabs do not have any under-coordinated atom relative to their bulk coordination in the first shell. If there is exactly one such (hkl), the material is a layered material with relatively large spatial gaps separating individual layers. If the number of directions is larger than one, then the corresponding systems are of lower dimensionality, quasi-1D for two such directions and molecular crystals for larger than two. If there are no such directions, the structure is then a connected 3D structure without large spatial gaps. In Ref. [192] we demonstrated the success of our algorithm in searching complex quasi-2D materials, including those with layer stacking in oblique directions, materials with corru- gated, accordion-like layers, and those with individual layers composed of multiple atomic layers. Using the described automated algorithm, we have in this work considered 3500 binary and 8000 additional ternary compounds from the ICSD and classified them into layered (quasi-2D) and not layered materials. We restricted our search to stoichiometric and ordered systems that do not contain rare earth elements and have 50 of less atoms in the unit cell. From our calculations, we have identified 426 binary and 443 ternary quasi-2D compounds. A full list of these materials can be found in supplementary information with their ICSD id.

4.2.2 Calculating Piezoelectric Properties of Quasi-2D Materials

In quasi-2D structures considered in this work, the individual layers are held together by relatively weak vdW interactions. The standard exchange-correlation functionals typically employed in density functional theory (DFT) calculations, including the calculations of elastic and piezoelectric properties, are known to fail to describe the vdW interactions. This is evident from relatively large errors in the out-of-plane lattice constants and the associated elastic properties [196]. To overcome this issue, we employed a vdW-corrected functional (optB86) as implemented in VASP (Vienna Ab-initio Simulation Package) code [197, 198] to calculate the lattice parameters, elastic, and piezoelectric properties of quasi-2D materials [199, 200]. To evaluate the piezoelectric coeﬃcient tensors, we utilize the VASP implemen-

71 ino lsi esr,adpeolcrccecettenso n coefficient piezoelectric for and used tensors, is elastic eV of 540 tion of energy cutoff wave plane large (DFPT) relatively theory perturbation functional density the of tation measure informations. with supplementary techniques, the all measurement in are of w provided functional Details compared vdW-corrected are measurements. values the calculated with The calcula calculations materials quasi-2D functional. of corrected constants Elastic 4.3: Figure atoms × n kpoints ≈ 00 where 1000, 1000 Experimental Cij (GPa) 1000 Experimental Cij (GPa) 100 100 10 10 1 1 MoS n (b) GGA+vdW (a) GGA 0101000 100 10 1 0101000 100 10 1 atoms HgI SnS 2 2 2 TiSe TaSe HgI Calculated Calculated snme faosi h rmtv eland cell primitive the in atoms of number is 50% error 2 2 2 50% error Bi BiI Bi 2 BiI 3 Te 72 2 TaSe Te SnS 3 3 3 2 2 TiSe C C ij ij (GPa) (GPa) SnS 2 MoS 2 s es -on rd endby defined grid, k-point dense A rs. 2 e ih()GAad()vdW- (b) and GGA (a) with ted C C C C C C C C 21 0,23 acltos A calculations. 203] 202, [201, 44 33 12 11 tutrlrlxto,calcula- relaxation, structural ettemperatures, ment n5%errrltv othe to relative error 50% in t xeietloe.The ones. experimental ith 44 33 12 11 etc. n kpoints are , input structures from ICSD database

Cij , eij C , e DFPT transition elements ij ij calculations present? quasi-2D? No Yes Cij Yes Yes eigenvalues of magnetic Cij , eij non- d C > 0? ground state? ij centrosymmetric? ij No

Yes Cij Yes Yes finite difference finite diff. and DFPT calc. Cij , eij relaxation E >0.1eV? g calculations at magnetic ground state

Figure 4.4: Workflow for calculations of piezoelectric coefficient tensor eij, the elastic stiffness Cij, and the piezoelectric modulus tensor dij of quasi-2D materials. A vdW-corrected functional (optB86) is used in all calculations. We have calculated piezoelectric modulus of 80 quasi-2D materials. is the number of k-points, is employed. In all our calculations, a very high tolerance of 10−8 eV for energy convergence is used, which is an important consideration for conducting DFPT calculations [177]. For calculation of elastic tensors we use a finite difference method. Here, the full elastic tensor is calculated by conducting six finite distortions of the lattice and obtaining elastic constants (Cij) from the stress-strain relationship [204, 205]. The importance of incorporating vdW-corrections is illustrated in Figure 4.3, where we notice significant improvement in predicting elastic constants, particularly C33 and C44, with vdW-corrected functional (optB86) [199, 200] compared to standard GGA-PBE functional [130]. The GGA-PBE predicted elastic constants are sourced from Ref.[11]. A more detailed analysis of the data presented in Figure 4.3 reveals that the GGA-PBE is still better in predicting in-plane elastic coefficients C11 and C12, but the error in reproducing C33 and

C44 is a factor of 10 or larger. This is due the fact that these particular two elastic constants are directly related to the deformations of the spatial gaps between the layers and the failure of GGA-PBE in reproducing relatively weak vdW interactions. The comparison of predicted properties with GGA-PBE and vdW-corrected functional is limited only to the elastic constants (Figure 4.3); the lack of experimental data on piezoelectric properties of

73 quasi-2D materials prevented us from making similar comparisons for predicted piezoelectric properties. In Table F.1 of Appendix F, we have provided the experimental details (e.g. measurement techniques, measuring temperatures, etc.) for each compound shown in Fig- ure 4.3. The comparison of calculated and experimental values for the piezoelectric modulus

of a few commercially important piezoelectric materials (including AlN and PbTiO3) are shown supplementary information. Predicted piezoelectric moduli are found to be in good agreement with experimental values.

4.2.3 Workﬂow for identifying quasi-2D piezoelectrics

A complete workflow we developed for identifying promising quasi-2D piezoelectric materials is illustrated in Figure 4.4. The binary and ternary crystal structures from the ICSD database are first screened using the automated algorithm for identifying quasi-2D structures. Then, we filter out all centrosymmetric structures based on the space group assigned in ICSD. Out of 11500 binary and ternary materials we find 869 layered systems, out of which 135 ∼ are identified as having non-centrosymmetric structures. Next, the non-centrosymmetric, structures are relaxed using the previously described first-principles calculations employing a vdW-corrected functional. As the piezoelectric materials need to have sizable band gaps for their properties to not be screened by the existence of free charge carriers we next employ a band-gap filter. As suggested in the previous works, [192, 206] for electronic structure calculation, we perform self-consistent GGA-PBE calculations on the vdW-relaxed structures using dense k-point grids. Because of the known band gap error in DFT calculations we use a relatively generous band gap cutoff of 0.1 eV. Fifty additional materials with their band gap smaller than 0.1 eV are discarded as a result. Finite difference calculations are performed with vdW-corrected functional to obtain the elastic constants (C). Five materials with elastic tensors with negative eigenvalues are also discarded. According to Born stability criteria,[207] the elastically stable materials always have positive eigenvalues of stiffness matrix. This means that an elastically stable materials always have positive elastic energy for arbitrary homogeneous deformation by an infinitesimal strain [208]. With the remaining 80

74 In2Te5 GeTe O PbS PbTiO3 100 CuVO In Te Sn 3 SnO NaI3O8 KSn2F5 TlBrO3 NaSnN Bi2WO6 In Te BiITe 2 5 SnO

(pC/N) Bi2O3 NaSnP SbF2Cl3 Ge max

d 10 SnS Cu2WS4 2 O F2NP AlN Te InSe GaInS Al2CdCl8 3 RhF3 Cu ZrCl2 V 1 1 36 160 186 230 GeTe CuVO Space Group 3

Figure 4.5: Left panel: plot of d = max( d ) against the space group number for 63 max | ij| quasi-2D vdW materials with dmax > 0 (out of 135 non-centrosymmetric compounds). The two horizontal dotted lines denote the d33 values of PbTiO3 (119 pC/N) and AlN (5.5 pC/N). The blue square in this plot represents the hypothetical structure of PbS (distorted NiAs found in ICSD), not the ground rocksalt structure. Right panel: crystal structures of quasi- 2D materials with large piezoelectric moduli. candidates, we proceed our calculations by performing Density Functional Perturbation The- ory (DFPT) [201, 202, 203] calculations to assess their piezoelectric coefficient tensor, e. Out of the remaining 80 candidates, 38 materials contain transition metal elements. For these, we perform a limited search of the magnetic ground state by enumerating all magnetic configurations in the unit cell and calculating their total energies. The elastic and piezo calculations are then performed only on the lowest energy spin state. This step is necessary because of the strong dependence of the electronic structure and other properties on spin configuration discussed in more details in Ref. [209]. The automated DFT calculations including initial file generation, calculating properties, data extraction and data handling are performed with the help of PyLada, [210] a Python framework for high-throughput first-principles calculations.

4.3 Results and Discussions 4.3.1 Promising Quasi-2D Piezoelectric Materials

Based on the calculated piezoelectric modulus tensor, a number of candidate materials with relatively large dij components have emerged. They are shown in Figure 4.5 with the full list together with the corresponding dij, Cij and eij values provided in the supplementary

75 information. In addition, the top 20 most promising systems, based on the largest dij component, are listed in Table 4.1. The piezoelectric modulus tensor (d) is a third rank tensor, and any isotropic averaging scheme will yield zero [211]. To rank these materials based on the merit of their piezoelectric response, we deﬁne d = max( d ) as the largest element of the absolute d matrix. max | ij| ij

Then the dmax is plotted against the space group number in Figure 4.5. The two reference lines have been drawn for categorizing these candidates – one representing calculated d33 of

PbTiO3 and the other representing calculated d33 of AlN (see supplementary information for the benchmark against experiments).

Please note that PbTiO3 is also a ferroelectric and here we are only using the value for

its piezoelectric response. More precisely, the PbTiO3 reference line corresponds to the bulk piezoelectric modulus corresponding to the single-crystal single-domain samples. The materials shown in the left panel of Figure 4.5 can broadly be divided in three

categories. The ﬁrst category is comprised of quasi-2D compounds with dmax larger than the

longitudinal piezoelectric modulus of PbTiO3 (d33=119 pC/N) [81] – the key end member

of most commercial high-strain piezoelectrics. We found three materials (In2Te5, PbS, and GeTe) in this category. Among them PbS is not in its ground state rocksalt phase, but in the hypothetical distorted NiAs structure which has found its way into the ICSD [212]. The other two compounds have previously been experimentally synthesized, [213, 214] but their piezoelectric moduli have not been reported so far.

In the second category we group all compounds which have dmax larger than the longi-

tudinal piezoelectric modulus of AlN (d33=5.5 pC/N) [78] and lower than the longitudinal

piezoelectric modulus of PbTiO3. The majority (48 compounds) of the piezoelectric candidates from our study fall in this category revealing that overall the vdW bonded quasi-2D systems indeed exhibit a propensity toward large piezo-response. This group is composed

from oxides and other chalcogenides such as CuVO3, SnO, BiInO3, MoV2O8, SnS2, InSe,

Cs2Te3 and other (total of 30); halides such as KSn2F5, AgI, MgCl2, and PbI2 (total of 4);

76 Table 4.1: List of top 20 candidate quasi-2D piezoelectric materials are shown with their space group (SG) number, calculated DFT band gap (Eg), maximal piezoelectric modulus dmax, the dij component that appears as dmax, maximal eij (emax), the emax component of eij.

2 2 Compound SG Eg(eV) dmax (pC/N) max dij emax (C/m ) max eij edmax (C/m ) In2Te5 9 0.7 351.7 d15 2.6 e15 2.6 † PbS 186 0.2 161.4 d33 8.3 e33 8.3 GeTe 160 0.6 148.4 d15 3.3 e15 3.3 CuVO3 1 0.9 106.9 d22 0.8 e32 0.2 SnO 31 1.6 67.1 d22 1.1 e22 1.1 BiInO3 33 2.8 56.1 d33 4.7 e33 4.7 ‡ Bi2WO6 41 1.7 54.1 d24 3.9 e33 2.9 NaI3O8 81 2.8 48.4 d14 0.7 e31 0.6 NaN3 12 1.4 40.7 d36 0.3 e34 0.1 KSn2F5 143 3.0 40.5 d15 0.3 e15 0.3 MoV2O8 35 0.8 40.4 d33 2.9 e33 2.9 TlBrO3 160 3.0 38.8 d24 1.0 e24 1.0 NaSnN 186 1.1 36.6 d15 0.6 e15 0.6 ‡ Cs2Te3 36 0.5 31.3 d36 0.6 e11 0.4 Bi2MoO6 61 1.7 28.7 d26 1.6 e11 1.3 AgI 186 1.3 27.8 d15 0.3 e33 0.1 SbF2Cl3 79 1.5 24.4 d15 0.2 e33 0.1 MgCl2 115 4.9 23.5 d15 0.2 e15 0.2 PbRb2O3 36 1.3 22.2 d34 0.7 e34 0.7 BiGeO5 9 2.3 21.1 d33 3.0 e33 3.0 † hypothetical structures ‡ dynamically unstable

77 pnictides such as NaSnN, NaSnP, KSnAs, and NaN3 (total of 4). Also, a number of materials

in this group (10) are the mixed anion systems, e.g., NaI3O8, SbF2Cl3. Not surprisingly, the more ionic systems like oxides, halides and nitrides are more frequently found closer to the top of the range. Finally, two compounds Cs2Te3 and Bi2WO6 are found in calculations to be dynamically unstable, but both have been experimentally synthesized (likely high-temperature phases) [215, 216]. The last group is composed of materials with the piezoelectric response lower than the longitudinal piezoelectric modulus of AlN. Though these candidates exhibit low piezoelectric response, they could still be useful as the calculated moduli are comparable with that

of quartz (d11=2.27 pC/N) [217]. We found a total of 12 compounds which fall in this cat-

egory. Examples include: WS2, RhF3, ZrCl2, and GaInS3. The moduli d, e and C with other informations such as band gap, space group of these compounds are provided in the supplementary information. We also observe that the piezoelectric compounds in the quasi-2D family of solids are

clustered mainly in three speciﬁc space groups, i.e., space group no. 36 (Cmc21), 160 (R3m),

and 186 (P63mc). This is mainly a reﬂection of the population bias, as these are the three most frequently occurring non-centrosymmetric space groups in the quasi-2D family of crystals.

Table ?? shows that in 9 of the high-response quasi-2D piezoelectric compounds, the d15

component appears as dmax. Note that because of the freedom in choosing the in-plane axes,

d15 and d24 are virtually indistinguishable (see the discussion section). This component corresponds to the thickness shearing deformation where the material shears like a deck of cards

in the in-plane direction, with no change in the other dimension. Materials with large d15 can be used in a variety of applications including: sensors, actuators, accelerometer, material testing structural health monitoring, non-destructive testing (NDT), and non-destructive

evaluation (NDE) [218]. The components of dij appearing as dmax usually coincide with the

components of eij appearing as emax. The distribution of diﬀerent components of dij (and

78 d ceaiso te eeatpeolcrcoeaigmode operating corres piezoelectric relevant their other with of schematics modes operating piezoelectric important iue46 itga ftemxmmcmoet f()tepiez the (a) of components maximum e the of Histogram 4.6: Figure e information. supplementary the in max ij max pern as appearing ) n b izeeti ouu tensor, modulus piezoelectric (b) and h otfeun aiu ausare values maximum frequent most the d max (and (b) (a) (c) (a) (b) (c)

V No. of Compounds No. of Compounds V 10 15 20 10 15 20 25 30 1 5 1 5 e max 123456222222 313233343536 212223242526 111213141516 313233343536 212223242526 111213141516 r icse nmr eali h etsection. next the in detail more in discussed are ) d d 16 35 ! V or or " or or ! Components of Components of " Components of Components of d d d ! 26 34 33 " d Components of Components of Components of Components of 1 max 79 3 d e h otfrequent most The . d 15 ij appearing as ij 2 appearing as , V d d e 24 ij ij and s, ! d " d odn eomto ye.The types. deformation ponding 15 e i.e. 14 d max V ! or or d max or or " d 36 d V 33 ! d opnnsof components , 24 " 25 c ceaiso several of Schematics (c) . eeti offiin tensor, coefficient oelectric e max is e 33 d hra for whereas , ij r shown are Another quantity that could influence the piezoelectric response is the band gap of the material. In this work the band gaps are calculated at the DFT level. Only about 1/3 of the studied materials are found to have DFT band gaps below 1 eV. Given the well known underestimation of band gaps in DFT based methods we do not think that materials with DFT band gaps larger than 1 eV would suffer from problems related to the existence of free charge carriers due to thermal fluctuations. However, for those with smaller gaps thermal fluctuation may cause sufficient number of free charge carriers, which may lower the polarization upon straining these materials despite having large piezoelectric moduli. Of course, provided that the real band gap is sufficiently close to the DFT one. For these materials, a more accurate assessment of the electronic structure might be needed before they are considered for applications. In relation to the chemical composition and toxicity it is also important to not that cur-

rently, the most widely used piezoelectric material is lead zirconate titanate (PbZr1−xTixO3 or PZT) [91]. However, PZT causes significant environmental problems because of its high lead content [91]. Hence, significant efforts have been made to develop lead-free piezoelectric materials [91, 73, 219]. In our work, we have identified 44 candidates that do not contain any toxic elements including Pb. Out of 44 candidates, 33 of them have their piezoelectric modulus larger than AlN.

4.3.2 Role of van der Waals Interactions

In order to understand the role of van der Waals interactions on the piezoelectric response

of quasi-2D materials, we analyze the relationships of emax and dmax to the corresponding strain and stress components, respectively. A histogram showing the number of compounds

with a given eij component appearing as emax is shown in Figure 4.6 (a). We observe that the most frequent emax is e33. This indicates that in the majority of quasi-2D materials the largest piezoelectric response, as measured by the eij, is along the layer stacking direction. The reason for this behavior is that the relatively weak vdW interactions allow large charge redistribution in the layering direction upon straining the system.

80 On the other hand, the piezoelectric modulus tensor d relates relates stress to polarization and combines two types of eﬀects: (1) amount of strain due to application of stress, and (2) amount of charge redistribution (polarization) due to resultant strain produced by the applied stress. A similar histogram of the dij components appearing as dmax is shown in

Figure 4.6(b). We divide the dij components into three groups depending on the deformation types (stress components) and the polarization direction. The schematics of the polarization directions and the associated stress components is shown in Figure 4.6(c) together with the dij components connecting the two. Every component of dij represents a separate piezoelectric operating mode. Schematics of all possible piezoelectric modes are provided in supplementary information. Group I: Applied stress deforms van der Waals bonds and the measured polarization coincides with the direction of the deformations. The longitudinal mode (d33) and shear modes (d15 and d24) fall in this class. As evident from the histogram in Figure 4.6(b) these are the most frequently appearing dmax components. In our considered materials, the modes d24 and d15 are indistinguishable because of the arbitrariness of the choice of axes ‘1’ and ‘2’, while the axis ‘3’ is ﬁxed by layer stacking directions. In both of these modes, the same stress component (σ4) is responsible for the deformation, which implies shearing of the van der Waals gaps. On the other hand, in the d33, the applied stress axially deforms (stretches or compresses) van der Waals gaps. Hence, the large piezoelectric responses are achieved by deforming (axial or shear) the relatively soft van der Waals bonds. The bar-heights of d15 and d24 in the histogram of dmax in Figure Figure 4.6(b) are larger compared to d33. This is because the shearing resistance values (C44) of quasi-2D materials are lower compared to their axial resistance values (C33) (refer to Figure 4.3(b)). Group II: Applied stress deforms van der Waals bonds, but the measuring polarization directions are diﬀerent from their deformation directions. The face shear modes (d14 and d25) and the thickness-extension modes (d31 and d32) fall in this class. The schematics of d31 and d32 are provided in the supplementary information. Usually, the direction of polarization is

81 facilitated by the direction of deformation. In such modes, the deformation directions are diﬀerent from their measured polarization directions. This is the reason behind their low

occurrence in the histogram of dmax though the van der Waals bond are deformed by the applied stress. Group III: The van der Waals bond does not deform by the applied stress in these modes.

This is why these modes do not appear frequently in the histogram of dmax. This includes length or width extension modes (d11 or d22) and shearing modes of type d16 and d36. The above discussion implies that the large piezoelectric response is always accompanied and caused by the stress that deforms the “soft” van der Waals gaps either through stretching, compression, or shearing. In addition, the analysis of the maximal piezoelectric moduli and the associated stress components provided guidance to experimentalists of how thin ﬁlms should be grown to utilize large dmax. It also describes what kind of mechanical actuation is necessary to achieve large piezoelectric response. This also helps in the design of new

devices to take advantage of large dmax. Finally, these ﬁndings can open up a wide variety of devices based on their operation modes or based on new materials for non-conventional

modes. For example, materials with large face-shear (d14) mode response will be attractive for for torsional applications, like novel gyroscopic sensors or high-precision torsional MEMS actuators [218].

4.3.3 Axial Piezoelectric and Elastic Anisotropy

In addition to revealing new candidate piezoelectric materials and explaining the origin of strong piezoelectric response, we have also investigated the anisotropy in axial piezoelectric response in relation to the axial elastic anisotropy. We analyze how the response in the out-of-plane direction compares to the in-plane responses. The in-plane and out-of-plane directions in quasi-2D materials are trivial to define and are illustrated in Figure 4.1. We define the axial anisotropy in both elastic and piezoelectric responses by the ratio of the out- of-plane component to the in-plane component. Here, the in-plane component is defined by the arithmetic mean of the ‘11’ and ‘22’-components (invariant to the choice of the in-plane

82 (a)

Bi2WO6 100 Al2ZnS4

) GeZnO 22 BN 3

d 10

+ NaN3 GeTe

11 WS 2 BiTeI d

/( 1 MoSe 2 Ag GeS 33 2 3 d 2 CuVO 0.1 3

Li7SbO6 0.01 0.01 0.1 1 10 100

2e33/(e11 + e22) (b)

Bi2WO6 100 Al2ZnS4 ) GeZnO3 22

d 10 BN BiTeI + GeTe NaN 11 WS2 3 d

/( 1 MoSe Ag2GeS3

33 2 d 2 CuVO 0.1 3

Li7SbO6 0.01 0.01 0.1 1 10 100

2C33/(C11 + C22)

Figure 4.7: Correlation between axial anisotropy, i.e., out-of-plane to in-plane response ratio) of (a) d and e (b) d and C. The results reveal that majority of quasi-2D piezoelectric materials are dominated by out-of-plane piezoelectric responses (both in e and d) but elastically they are dominated in in-plane direction.

axes), whereas the out-of-plane response is solely deﬁned by the ‘33’-component. Hence, the axial anisotropy of d, e, and C can be expressed as 2d33/(d11 + d22), 2e33/(e11 + e22), and 2C33/(C11 + C22) respectively. If the axial anisotropy equals or nearly equals 1 then the

83 material is considered to be isotropic in the corresponding quantity responses with respect to the in-plane and out-of-plane directions. If the axial anisotropy is greater (or lower) than 1 the response of a material to axial deformation is dominated in out-of-plane (in-plane) direction. All possible correlations among the axial anisotropy of d, e, and C have been investigated. The correlation between axial anisotropy of d with respect to the axial anisotropy of e and C are shown in Figure 4.7 (a) and (b) respectively. From the comparative studies between Figure 4.7 (a) and (b), we see that the axial anisotropy of d is mainly dictated by the axial anisotropy of e not by the axial anisotropy of C. The plot between axial anisotropy of C and e is provided in the supplementary information. From Figure 4.7 (a) and (b), we observe that most quasi-2D materials have axial piezoelectric anisotropy parameter (both in d and e) > 1 and axial elastic anisotropy parameter < 1. This implies that the quasi-2D piezoelectric materials are dominant in out-of-plane piezoelectric responses but elastically they are dominant in-plane directions. This result cor- roborates the correlation between elastic softness and large piezoelectric response, i.e., the large piezoelectric responses are observed in elastically softer directions. We found quasi-2D piezoelectric compounds such as CuVO3 and Li7SbO6 are dominated by in-plane piezoelectric response. Compounds such as Bi2WO6, Al2ZnS4, GeZnO3, and GeTe are nearly isotropic in their axial elastic responses but highly anisotropic in axial piezoelectric responses.

4.4 Conclusions

In conclusion, we performed a large-scale computational (ﬁrst-principles) assessment of the bulk piezoelectric properties of layered (quasi-2D), vdW bonded materials. In our study we concentrate on the piezoelectric modulus as the measure of the piezoelectric response, which relates mechanical stress and electric polarization and depends on a combination of charge redistribution due to strain and the amount of strain produced by the stress. Overall, out of 135 non-centrosymmetric quasi-2D binary and ternary structures from ICSD we have discovered 51 materials with piezoelectric response larger than that of AlN, a well-

84 known piezoelectric materials used in applications. Out of these 51 systems, we ﬁnd three

with the piezoelectric modulus even larger than that of PbTiO3 that has the piezoelectric modulus among the largest known. More importantly, 33 out of the 51 layered compounds do not contain any toxic elements including Pb. Our results also reveal that the large piezoelectric modulus in vdW systems is directly enabled by the vdW interactions between layers as in majority of compounds the large components of the piezoelectric modulus tensor couple to the stress components that imply deformations (both shear and axial) of the “soft” vdW bonds between layers. Our results suggest that quasi-2D layered materials are a rich structural space for discovering new piezoelectric materials.

85 CHAPTER 5 SUMMARY & CONCLUSIONS

In this chapter, we summarize the results of this thesis and highlight its key accom- plishments. In addition, we suggest a few future research directions that could provide the next steps towards the ultimate goal of designing novel piezoelectric materials with tailored physical properties.

5.1 Conclusions

This thesis is composed of these three works: (a) Improving piezoelectric response of AlN by alloying, (b) Tuning the piezoelectric and mechanical properties of AlN by co-alloying, and (c) Discovering new piezoelectric materials with large piezoelectric modulus By density functional theory, we conclude that (Cr, Al)N has a low wurtzite-to-rocksalt transition concentration, and therefore can oﬀer certain piezoelectric enhancements at alloying levels that are easier to stabilize during the synthesis. Among a set of wurtzite-structured materials, we have found that (Cr, Al)N has the most sensitive piezoelectric coeﬃcient with

respect to alloying concentration. Our results reveal that, for 12.5% Cr d33 is twice that of pure AlN, and for 30% Cr this modulus is about four times larger than that of AlN. This ﬁnding, combined with the relative ease of synthesis under non-equilibrium conditions, may establish (Cr,Al) N as a prime piezoelectric material for applications such as resonators and acoustic wave generators. Improvement of piezoelectric properties by transition metal alloying is also accompanied by elastic softening in AlN. Co-alloying of boron nitride and transition metal metal nitride such as YN with AlN oﬀers superior combination of both the mechanical and piezoelectric properties of AlN. For discovery of new piezoelectric materials, we perform high-throughput screening in layered binary and ternary materials for their piezoelectric properties. We discovered 50

86 new compounds such as In2Te5, GeTe, and CuVO3 with piezoelectric response larger than AlN. As a matter of fact, 70% of our discovered piezoelectric candidates do not contain any toxic elements like Pb. Our analysis revealed that the large components of d always couple with the shearing or axial deformations of van der Waals “gaps” between the layers.

5.2 Recommendations for Future Work

This dissertation focuses on designing and discovering new piezoelectric materials using density functional theory. Here, we list following two areas which can be explored in the future: (1) The effect of alloying at cationic site of AlN on piezoelectric and mechanical properties has been studied chapter 2 and chapter 3. The future work can be extended on the studying the effect of N-site alloying by alloying with P on piezoelectric properties of AlN. AlP has a low bulk modulus compared to AlN. The systematic studies on enthalpy of formation, T x − curve will also reveal the stability of Al(P,N) alloys which have not been explored previously. (2) In the initial investigations mentioned previously, we found several candidate materials with high piezoelectric response. Ferroelectrics are a subset of piezoelectrics with spontaneous polarization switchable by an applied electric field. In future investigations, one could identify the potential quasi-2D ferroelectric materials from this set of quasi-2D piezoelectric materials. Identification of ferroelectric materials with unique responses such as spontaneous polarization and polarization reversibility would befit their application in capacitors, non-volatile memory, and transpolarizers.

87 REFERENCES CITED

[1] Jack W Judy. Microelectromechanical systems (mems): fabrication, design and applications. Smart materials and Structures, 10(6):1115, 2001.

[2] Henry A Sodano, Daniel J Inman, and Gyuhae Park. A review of power harvesting from vibration using piezoelectric materials. Shock and Vibration Digest, 36(3):197– 206, 2004.

[3] Jiashi Yang. Piezoelectric transformer structural modeling-a review. IEEE transactions on ultrasonics, ferroelectrics, and frequency control, 54(6), 2007.

[4] MARKETSANDMARKET. Piezoelectric Devices Market worth 31.33 billion usd by 2022. https://www.marketsandmarkets.com/PressReleases/ piezoelectric-devices.asp, 2017.

[5] Walter Heywang, Karl Lubitz, and Wolfram Wersing. Piezoelectricity: evolution and future of a technology, volume 114. Springer Science & Business Media, 2008.

[6] Kenji Uchino. Piezoelectric ultrasonic motors: overview. Smart materials and structures, 7(3):273, 1998.

[7] K Uchino. The development of piezoelectric materials and the new perspective. In Advanced Piezoelectric Materials (Second Edition), pages 1–92. Elsevier, 2017.

[8] Steven R Anton and Henry A Sodano. A review of power harvesting using piezoelectric materials (2003–2006). Smart materials and Structures, 16(3):R1, 2007.

[9] Mahmoud Al Ahmad and Areen Allataifeh. Piezoelectric-based energy harvesting for smart city application. In Information Innovation Technology in Smart Cities, pages 343–356. Springer, 2018.

[10] J¨urgen Hafner, Christopher Wolverton, and Gerbrand Ceder. Toward computational materials design: the impact of density functional theory on materials research. MRS bulletin, 31(9):659–668, 2006.

88 [11] Anubhav Jain, Shyue Ping Ong, Geoﬀroy Hautier, Wei Chen, William Davidson Richards, Stephen Dacek, Shreyas Cholia, Dan Gunter, David Skinner, Gerbrand Ceder, and Kristin A. Persson. Commentary: The materials project: A materials genome approach to accelerating materials innovation. APL Materials, 1(1):011002, 2013.

[12] Prasanna V Balachandran, Antoine A Emery, James E Gubernatis, Turab Lookman, Chris Wolverton, and Alex Zunger. Predictions of new ab o3 perovskite compounds by combining machine learning and density functional theory. Physical Review Materials, 2(4):043802, 2018.

[13] Xiuwen Zhang, V Stevanovi´c, Mayeul dAvezac, Stephan Lany, and Alex Zunger. Pre- diction of a2bx4 metal-chalcogenide compounds via ﬁrst-principles thermodynamics. Physical Review B, 86(1):014109, 2012.

[14] K Uchino. Materials issues in design and performance of piezoelectric actuators: an overview. Acta Materialia, 46(11):3745–3753, 1998.

[15] John Frederick Nye. Physical properties of crystals: their representation by tensors and matrices. Oxford university press, 1985.

[16] Franco Jona and Gen Shirane. Ferroelectric crystals, volume 1. Pergamon, 1962.

[17] Bernard Jaﬀe. Piezoelectric ceramics, volume 3. Elsevier, 2012.

[18] Woldemar Voigt. Lehrbuch der kristallphysik (mit ausschluss der kristalloptik). Springer-Verlag, 2014.

[19] I Standard. Ieee standard on piezoelectricity. ANSI/IEEE Std, pages 176–1987, 1988.

[20] VZ Parton and BA Kudryavtsev. Electromagnetoelasticity. Gordon and Breach Science Publishers, New York, 2:90059–0, 1988.

[21] David S Eddy and Douglas R Sparks. Application of mems technology in automotive sensors and actuators. Proceedings of the IEEE, 86(8):1747–1755, 1998.

[22] Randy Frank. Understanding smart sensors. Artech House, 2013.

[23] Kenji Uchino. Piezoelectric actuators and ultrasonic motors, volume 1. Springer Science & Business Media, 1996.

89 [24] Dominik J Bell, TJ Lu, Norman A Fleck, and Simon M Spearing. Mems actuators and sensors: observations on their performance and selection for purpose. Journal of Micromechanics and Microengineering, 15(7):S153, 2005.

[25] QQ Zhang, SJ Gross, S Tadigadapa, TN Jackson, FT Djuth, and S Trolier-McKinstry. Lead zirconate titanate ﬁlms for d33 mode cantilever actuators. Sensors and Actuators A: Physical, 105(1):91–97, 2003.

[26] Marc Zupan, Mike F Ashby, and Norman A Fleck. Actuator classiﬁcation and selec- tionthe development of a database. Advanced Engineering Materials, 4(12):933–940, 2002.

[27] SM Spearing. Materials issues in microelectromechanical systems (mems). Acta materialia, 48(1):179–196, 2000.

[28] Ahmad Safari and E Koray Akdogan. Piezoelectric and acoustic materials for transducer applications. Springer Science & Business Media, 2008.

[29] Hiroshi Nakatsuka, Keiji Onishi, Hiroyuki Nakamura, and Takehiko Yamakawa. Piezo- electric resonator, ﬁlter, and duplexer, August 21 2007. US Patent 7,259,498.

[30] Robert Aigner. Thin ﬁlm piezoelectric resonator, December 2 2003. US Patent 6,657,363.

[31] Hirofumi Takahashi, Yoshiki Numamoto, Junji Tani, and Sadahiro Tsurekawa. Piezo- electric properties of BaTiO3ceramics with high performance fabricated by microwave sintering. Japanese Journal of Applied Physics, 45(9S):7405, 2006.

[32] Gianluca Piazza, Valeriy Felmetsger, Paul Muralt, Roy H Olsson III, and Richard Ruby. Piezoelectric aluminum nitride thin ﬁlms for microelectromechanical systems. MRS bulletin, 37(11):1051–1061, 2012.

[33] Hidekazu Kitamura, Kazuhiro Inoue, and Masaki Takeuchi. Piezoelectric resonator, August 27 2002. US Patent 6,441,539.

[34] Li-Wen Hung and Clark T-C Nguyen. Capacitive-piezoelectric aln resonators with q> 12,000. In Micro Electro Mechanical Systems (MEMS), 2011 IEEE 24th International Conference on, pages 173–176. IEEE, 2011.

90 [35] Ting-Ta Yen, Albert P Pisano, and Clark T-C Nguyen. High-q capacitive-piezoelectric aln lamb wave resonators. In Micro Electro Mechanical Systems (MEMS), 2013 IEEE 26th International Conference on, pages 114–117. IEEE, 2013.

[36] Chih-yi Lin. Design and analysis of piezoelectric transformer converters. PhD thesis, Virginia Tech, 1997.

[37] Paul Muralt. Piezoelectric thin ﬁlms for mems. Integrated Ferroelectrics, 17(1-4):297– 307, 1997.

[38] Shashank Priya and Daniel J Inman. Energy harvesting technologies, volume 21. Springer, 2009.

[39] A Ballato. Basic material quartz and related innovations. In Piezoelectricity, pages 9–35. Springer, 2008.

[40] Shujun Zhang and Fapeng Yu. Piezoelectric materials for high temperature sensors. Journal of the American Ceramic Society, 94(10):3153–3170, 2011.

[41] Franck LEVASSORT, L Pascal TRAN-HUU-HUE, Dominique CERTON, and Marc LETHIECQ. Piezoelectric materials for ultrasonic transducers: Review of recent developments, 2002.

[42] Dragan Damjanovic. Materials for high temperature piezoelectric transducers. Current Opinion in Solid State and Materials Science, 3(5):469–473, 1998.

[43] Luis AO Araujo, Cesar R Foschini, and Carlos A Fortulan. Piezoelectric energy harvesting device based on quartz as a power generator. In Magnetic, Ferroelectric, and Multiferroic Metal Oxides, pages 265–274. Elsevier, 2018.

[44] Y Saigusa. Quartz-based piezoelectric materials. In Advanced Piezoelectric Materials (Second Edition), pages 197–233. Elsevier, 2017.

[45] SJZH ANG, JL UO, and TRSHR OUT. High-performance, high-Tc piezoelectric crystals. Handbook of Advanced Dielectric, Piezoelectric and Ferroelectric Materials: Syn- thesis, Properties and Applications, page 130, 2008.

[46] P Krempl, G Schleinzer, W Wallno, et al. Gallium phosphate, GaPO4: a new piezoelectric crystal material for high-temperature sensorics. Sensors and Actuators A: Physical, 61(1-3):361–363, 1997.

91 [47] Jeﬀrey L Bleustein. A new surface wave in piezoelectric materials. Applied Physics Letters, 13(12):412–413, 1968.

[48] Susan Trolier-McKinstry, Shujun Zhang, Andrew J Bell, and Xiaoli Tan. High- performance piezoelectric crystals, ceramics, and ﬁlms. Annual Review of Materials Research, (0), 2017.

[49] Renee L Bunde, Eric J Jarvi, and Jeﬀrey J Rosentreter. Piezoelectric quartz crystal biosensors. Talanta, 46(6):1223–1236, 1998.

[50] R Bechmann. Elastic and piezoelectric constants of α-quartz. Physical Review, 110(5):1060, 1958.

[51] quartz1. Quartz watch. http://resonanceswavesandfields.blogspot.co.uk/ 2011/03/understanding-quarz-analog-mechanical.html, 2013.

[52] RS Weis and TK Gaylord. Lithium niobate: summary of physical properties and crystal structure. Applied Physics A, 37(4):191–203, 1985.

[53] Wang Yue and Jiang Yi-Jian. Crystal orientation dependence of piezoelectric properties in LiNbO3 and LiTaO3. Optical Materials, 23(1-2):403–408, 2003.

[54] Single crystal FBAR with LiNbO3 and LiTaO]3 piezoelectric substance layers, author=Osugi, Yukihisa and Yoshino, Takashi and Suzuki, Kenji and Hirai, Takami, booktitle=Microwave Symposium, 2007. IEEE/MTT-S International, pages=873–876, year=2007, organization=IEEE.

[55] Albert A Ballman. Growth of piezoelectric and ferroelectric materials by the czochraiski technique. Journal of the American Ceramic Society, 48(2):112–113, 1965.

[56] Il Hyoung Jung and Keun Ho Auh. Crystal growth and piezoelectric properties of langasite (La3Ga5SiO14) crystals. Materials Letters, 41(5):241–246, 1999.

[57] J Stade, L Bohat`y, M Hengst, and RB Heimann. Electro-optic, piezoelectric and dielectric properties of Langasite (La3Ga5Sio14), Langanite (La3Ga5.5Nb0.5O14) and Lan- gataite (La3Ga5.5Ta0.5O14). Crystal Research and Technology: Journal of Experimental and Industrial Crystallography, 37(10):1113–1120, 2002.

[58] T Iwataki, H Ohsato, K Tanaka, H Morikoshi, J Sato, and K Kawasaki. Mechanism of the piezoelectricity of langasite based on the crystal structures. Journal of the European Ceramic Society, 21(10-11):1409–1412, 2001.

92 [59] J Bohm, RB Heimann, M Hengst, R Roewer, and J Schindler. Czochralski growth and characterization of piezoelectric single crystals with langasite structure: La3Ga5SiO14 (lgs), La3Ga5.5Nb0.5O14 (LGN), and La3Ga5.5Ta0.5O14 (LGT): Part I. Journal of Crys- tal Growth, 204(1-2):128–136, 1999.

[60] Paul Schiopu, Ionica Cristea, Neculai Grosu, and Alexandru Craciun. Development of saw ﬁlters based on GaPO4. In Design and Technology in Electronic Packaging (SIITME), 2011 IEEE 17th International Symposium for, pages 169–172. IEEE, 2011.

[61] O Le Traon, S Masson, C Chartier, and D Janiaud. LGS and GaPO4 piezoelectric crystals: New results. Solid State Sciences, 12(3):318–324, 2010.

[62] Christian Reiter, Peter W Krempl, Herbert Thanner, Wolfgang Walln¨ofer, and Peter M Worsch. Material properties of GaPO4 and their relevance for applications. In Annales de chimie science des mat´eriaux, volume 26, pages 91–94. Elsevier, 2001.

[63] Manhal Souleiman, GM Bhalerao, T Guillet, Abel Haidoux, Martine Cambon, Claire Levelut, Julien Haines, and Olivier Cambon. Hydrothermal growth of large piezoelectric single crystals of GaAsO4. Journal of Crystal Growth, 397:29–38, 2014.

[64] L Eric Cross. Relaxor ferroelectrics. Ferroelectrics, 76(1):241–267, 1987.

[65] Seung-Eek Park and Thomas R Shrout. Relaxor based ferroelectric single crystals for electro-mechanical actuators. Materials Research Innovations, 1(1):20–25, 1997.

[66] MK Durbin, EW Jacobs, JC Hicks, and S-E Park. In situ x-ray diﬀraction study of an electric ﬁeld induced phase transition in the single crystal relaxor ferroelectric, 92% Pb(Zn1/3Nb 2/3)O3-8%PbTiO3. Applied Physics Letters, 74(19):2848–2850, 1999.

[67] Zhi-Wen Yin, Hao-Su Luo, Ping-Chu Wang, and Gui-Sheng Xu. Growth, characterization and properties of relaxor ferroelectric PMN-PT single crystals. Ferroelectrics, 229(1):207–216, 1999.

[68] Yoichi Kizaki, Yuji Noguchi, and Masaru Miyayama. Defect control for low leakage current in K0.5Na0.5NbO3 single crystals. Applied Physics Letters, 89(14):142910, 2006.

[69] Wenfeng Liu and Xiaobing Ren. Large piezoelectric eﬀect in Pb-free ceramics. Physical Review Letters, 103(25):257602, 2009.

93 [70] Masato Matsubara, Toshiaki Yamaguchi, Wataru Sakamoto, Koichi Kikuta, Toshi- nobu Yogo, and Shin-ichi Hirano. Processing and piezoelectric properties of lead-free (K,Na)(Nb, Ta) O3 ceramics. Journal of the American Ceramic Society, 88(5):1190– 1196, 2005.

[71] Susan Trolier-McKinstry and Peter Muralt. Thin ﬁlm piezoelectrics for MEMS. Journal of Electroceramics, 12(1-2):7–17, 2004.

[72] Dragan Damjanovic. Ferroelectric, dielectric and piezoelectric properties of ferroelectric thin ﬁlms and ceramics. Reports on Progress in Physics, 61(9):1267, 1998.

[73] Tadashi Takenaka and Hajime Nagata. Current status and prospects of lead-free piezoelectric ceramics. Journal of the European Ceramic Society, 25(12):2693–2700, 2005.

[74] T Takenaka, H Nagata, Y Hiruma, Y Yoshii, and K Matumoto. Lead-free piezoelectric ceramics based on perovskite structures. Journal of Electroceramics, 19(4):259–265, 2007.

[75] C-S Tu, IG Siny, and V Hugo Schmidt. Sequence of dielectric anomalies and high- temperature relaxation behavior in Na1/2Bi1/2TiO3. Physical Review B, 49(17):11550, 1994.

[76] Morito Akiyama, Toshihiro Kamohara, Kazuhiko Kano, Akihiko Teshigahara, Yukihiro Takeuchi, and Nobuaki Kawahara. Enhancement of piezoelectric response in scandium aluminum nitride alloy thin ﬁlms prepared by dual reactive cosputtering. Advanced Materials, 21(5):593–596, 2009.

[77] YB Jeon, R Sood, J-H Jeong, and S-G Kim. Mems power generator with transverse mode thin ﬁlm pzt. Sensors and Actuators A: Physical, 122(1):16–22, 2005.

[78] Marc-Alexandre Dubois and Paul Muralt. Properties of aluminum nitride thin ﬁlms for piezoelectric transducers and microwave ﬁlter applications. Applied Physics Letters, 74(20):3032–3034, 1999.

[79] AE Crawford. Lead zirconate-titanate piezoelectric ceramics. British Journal of Ap- plied Physics, 12(10):529, 1961.

[80] X Ming Lu and TL Proulx. Single crystals vs. PZT ceramics for medical ultrasound applications. In Ultrasonics Symposium, 2005 IEEE, volume 1, pages 227–230. IEEE, 2005.

94 [81] B. Jaﬀe. Piezoelectric Ceramics. Non-Metallic Solids. Elsevier Science, 2012.

[82] M Acosta, N Novak, V Rojas, S Patel, R Vaish, J Koruza, GA Rossetti Jr, and J R¨odel. BaTiO3-based piezoelectrics: Fundamentals, current status, and perspectives. Applied Physics Reviews, 4(4):041305, 2017.

[83] PK Panda and B Sahoo. PZT to lead free piezo ceramics: a review. Ferroelectrics, 474(1):128–143, 2015.

[84] Yong-hyun Lee, Jeong-ho Cho, Byung-ik Kim, and Duck-kyun Choi. Piezoelectric properties and densiﬁcation based on control of volatile mass of potassium and sodium in K0.5Na0.5NbO3 ceramics. Japanese Journal of Applied Physics, 47(6R):4620, 2008.

[85] Daoli Wang, Kongjun Zhu, Hongli Ji, and Jinhao Qiu. Two-step sintering of the pure K0.5Na0.5NbO3 lead-free piezoceramics and its piezoelectric properties. Ferroelectrics, 392(1):120–126, 2009.

[86] Yiping Guo, Ken-ichi Kakimoto, and Hitoshi Ohsato. (Na0.5K0.5)NbO3-LiTaO3 lead- free piezoelectric ceramics. Materials Letters, 59(2-3):241–244, 2005.

[87] T Jardiel, AC Caballero, and M Villegas. Aurivillius ceramics: Bi4Ti3O12-based piezoelectrics. Journal of the Ceramic Society of Japan, 116(1352):511–518, 2008.

[88] Alberto Moure, Alicia Castro, and Lorena Pardo. Aurivillius-type ceramics, a class of high temperature piezoelectric materials: Drawbacks, advantages and trends. Progress in Solid State Chemistry, 37(1):15–39, 2009.

[89] NW Emanetoglu, C Gorla, Y Liu, S Liang, and Y Lu. Epitaxial ZnO piezoelectric thin ﬁlms for saw ﬁlters. Materials Science in Semiconductor Processing, 2(3):247–252, 1999.

[90] M Demartin Maeder, D Damjanovic, and N Setter. Lead free piezoelectric materials. Journal of Electroceramics, 13(1-3):385–392, 2004.

[91] Thomas R Shrout and Shujun J Zhang. Lead-free piezoelectric ceramics: Alternatives for PZT? Journal of Electroceramics, 19(1):113–126, 2007.

[92] Thomas Oliver. Lead poisoning. PB Hoeber, 1914.

95 [93] Stephen Hall. Lead pollution and poisoning. Environmental Science & Technology, 6(1):30–35, 1972.

[94] Supriyo Ghosh. Predictive modeling of solidiﬁcation during laser additive manufacturing of nickel superalloys: recent developments, future directions. Materials Research Express, 5(1):012001, 2018.

[95] Supriyo Ghosh, Li Ma, Lyle E Levine, Richard E Ricker, Mark R Stoudt, Jarred C Heigel, and Jonathan E Guyer. Single-track melt-pool measurements and microstruc- tures in inconel 625. JOM, pages 1–6, 2018.

[96] Tamoghna Chakrabarti, Nisha Verma, and Sukriti Manna. Grain boundary driven plateau–rayleigh instability in multilayer nanocrystalline thin ﬁlm: A phase-ﬁeld study. Materials & Design, 119:425–436, 2017.

[97] Tamoghna Chakrabarti and Sukriti Manna. Zener pinning through coherent precipi- tates: A phase ﬁeld study. arXiv preprint arXiv:1805.03831, 2018.

[98] Supriyo Ghosh, Li Ma, Nana Ofori-Opoku, and Jonathan E Guyer. On the primary spacing and microsegregation of cellular dendrites in laser deposited ni–nb alloys. Mod- elling and simulation in materials science and engineering, 25(6):065002, 2017.

[99] Supriyo Ghosh, Nana Ofori-Opoku, and Jonathan E Guyer. Simulation and analysis of γ-ni cellular growth during laser powder deposition of ni-based superalloys. Com- putational Materials Science, 144:256–264, 2018.

[100] Toshio Kamiya. Calculation of crystal structures, dielectric constants and piezoelectric properties of wurtzite-type crystals using ab-initio periodic Hartree-Fock method. Japanese Journal of Applied Physics, 35(8R):4421, 1996.

[101] Fabio Bernardini, Vincenzo Fiorentini, and David Vanderbilt. Spontaneous polarization and piezoelectric constants of iii-v nitrides. Physical Review B, 56(16):R10024, 1997.

[102] Ferenc Tasnadi, Bj¨orn Alling, Carina H¨oglund, Gunilla Wingqvist, Jens Birch, Lars Hultman, and Igor A Abrikosov. Origin of the anomalous piezoelectric response in wurtzite ScxAl1−xN alloys. Physical review letters, 104(13):137601, 2010.

96 [103] Piezoelectric coeﬃcients and spontaneous polarization of ScAlN, author=Caro, Miguel A and Zhang, Siyuan and Riekkinen, Tommi and Ylilammi, Markku and Moram, Michelle A and Lopez-Acevedo, Olga and Molarius, Jyrki and Laurila, Tomi, journal=Journal of Physics: Condensed Matter, volume=27, number=24, pages=245901, year=2015, publisher=IOP Publishing.

[104] Sukriti Manna, Kevin R. Talley, Prashun Gorai, John Mangum, Andriy Zakutayev, Geoﬀ L. Brennecka, Vladan Stevanovi´c, and Cristian V. Ciobanu. Enhanced piezoelectric response of AlN via CrN alloying. Phys. Rev. Applied, 9:034026, Mar 2018.

[105] Sukriti Manna, Geoﬀ L Brennecka, Vladan Stevanovi´c, and Cristian V Ciobanu. Tun- ing the piezoelectric and mechanical properties of the AlN system via alloying with YN and BN. Journal of Applied Physics, 122(10):105101, 2017.

[106] Yong Qing Fu, JK Luo, Nam-Trung Nguyen, AJ Walton, Andrew John Flewitt, Xiao- Tao Zu, Yifan Li, Glen McHale, Allan Matthews, Enrique Iborra, et al. Advances in piezoelectric thin ﬁlms for acoustic biosensors, acoustoﬂuidics and lab-on-chip applications. Progress in Materials Science, 89:31–91, 2017.

[107] Paul Muralt. Recent progress in materials issues for piezoelectric MEMS. Journal of the American Ceramic Society, 91(5):1385–1396, 2008.

[108] HP Loebl, M Klee, C Metzmacher, W Brand, R Milsom, and P Lok. Piezoelectric thin aln ﬁlms for bulk acoustic wave (baw) resonators. Materials Chemistry and Physics, 79(2-3):143–146, 2003.

[109] Roy H OlssonIII, Kenneth E Wojciechowski, Michael S Baker, Melanie R Tuck, and James G Fleming. Post-cmos-compatible aluminum nitride resonant mems accelerometers. Journal of Microelectromechanical Systems, 18(3):671–678, 2009.

[110] Yixiang Wang, Hong Ding, Xianhao Le, Wen Wang, and Jin Xie. A MEMS piezoelectric in-plane resonant accelerometer based on aluminum nitride with two-stage microleverage mechanism. Sensors and Actuators A: Physical, 254:126–133, 2017.

[111] Chengjie Zuo, Jan Van der Spiegel, and Gianluca Piazza. 1.05-GHz CMOS oscillator based on lateral-ﬁeld-excited piezoelectric AlN contour-mode MEMS resonators. IEEE transactions on ultrasonics, ferroelectrics, and frequency control, 57(1), 2010.

97 [112] JL Fu, Y Nakano, LD Sorenson, and F Ayazi. Multi-axis aln-on-silicon vibration energy harvester with integrated frequency-upconverting transducers. In Micro Electro Mechanical Systems (MEMS), 2012 IEEE 25th International Conference on, pages 1269–1272. IEEE, 2012.

[113] R Elfrink, TM Kamel, M Goedbloed, S Matova, D Hohlfeld, Y Van Andel, and R Van Schaijk. Vibration energy harvesting with aluminum nitride-based piezoelectric devices. Journal of Micromechanics and Microengineering, 19(9):094005, 2009.

[114] C-M Yang, K Uehara, S-K Kim, S Kameda, H Nakase, and K Tsubouchi. Highly c-axis-oriented aln ﬁlm using mocvd for 5ghz-band fbar ﬁlter. In Ultrasonics, 2003 IEEE Symposium on, volume 1, pages 170–173. IEEE, 2003.

[115] Paul Muralt. Aln thin ﬁlm processing and basic properties. In Piezoelectric MEMS Resonators, pages 3–37. Springer, 2017.

[116] By convention, we refer to the eij values, which couple strain and dielectric displacement (or electric field and stress), as piezoelectric coefficients, while the dij values, which couple strain with electric field (or stress with dielectric displacement), are referred to as piezoelectric moduli. in both cases, we express the individual matrix elements in standard reduced voigt notation.

[117] Xiaoyu Kang, Smitha Shetty, Lauren Garten, Jon F Ihlefeld, Susan Trolier-McKinstry, and Jon-Paul Maria. Enhanced dielectric and piezoelectric responses in Zn1−xMgxO thin ﬁlms near the phase separation boundary. Applied Physics Letters, 110(4):042903, 2017.

[118] Carina Höglund, Jens Birch, Björn Alling, Javier Bareño, Zsolt Czigány, Per OA˚ Pers- son, Gunilla Wingqvist, Agne Zukauskaite, and Lars Hultman. Wurtzite structure sc 1- x al x n solid solution films grown by reactive magnetron sputter epitaxy: Struc- tural characterization and first-principles calculations. Journal of Applied Physics, 107(12):123515, 2010.

[119] PM Mayrhofer, H Riedl, H Euchner, M St¨oger-Pollach, PH Mayrhofer, A Bittner, and U Schmid. Microstructure and piezoelectric response of YxAl1−xN thin ﬁlms. Acta Materialia, 100:81–89, 2015.

[120] PH Mayrhofer, D Music, Th Reeswinkel, H-G Fuß, and JM Schneider. Structure, elastic properties and phase stability of cr1–xalxn. Acta Materialia, 56(11):2469–2475, 2008.

98 [121] David Holec, Florian Rovere, Paul H Mayrhofer, and P´eter B Barna. Pressure- dependent stability of cubic and wurtzite phases within the tin–aln and crn–aln systems. Scripta Materialia, 62(6):349–352, 2010.

[122] Alex Zunger, S-H Wei, LG Ferreira, and James E Bernard. Special quasirandom structures. Physical Review Letters, 65(3):353, 1990.

[123] Axel van de Walle. Multicomponent multisublattice alloys, nonconﬁgurational entropy and other additions to the alloy theoretic automated toolkit. Calphad, 33(2):266–278, 2009.

[124] A Van de Walle, P Tiwary, M De Jong, DL Olmsted, M Asta, A Dick, D Shin, Y Wang, L-Q Chen, and Z-K Liu. Eﬃcient stochastic generation of special quasirandom structures. Calphad, 42:13–18, 2013.

[125] Yasushi Endo, Takanobu Sato, Yoshio Kawamura, and Masahiko Yamamoto. Crystal structure and magnetic properties of cr-doped aln ﬁlms with various cr concentrations. Materials transactions, 48(3):465–470, 2007.

[126] Yasushi Endo, Takanobu Sato, Ayumu Takita, Yoshio Kawamura, and Masahiko Ya- mamoto. Magnetic, electrical properties, and structure of cr-aln and mn-aln thin ﬁlms grown on si substrates. IEEE transactions on Magnetics, 41(10):2718–2720, 2005.

[127] Bj¨orn Alling, Tobias Marten, and IA Abrikosov. Eﬀect of magnetic disorder and strong electron correlations on the thermodynamics of crn. Physical Review B, 82(18):184430, 2010.

[128] Igor A Abrikosov, AV Ponomareva, Peter Steneteg, SA Barannikova, and Bj¨orn Alling. Recent progress in simulations of the paramagnetic state of magnetic materials. Cur- rent Opinion in Solid State and Materials Science, 20(2):85–106, 2016.

[129] Georg Kresse and Jürgen Furthmüller. Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set. Computational materials science, 6(1):15–50, 1996.

[130] John P Perdew, Kieron Burke, and Matthias Ernzerhof. Generalized gradient approximation made simple. Physical review letters, 77(18):3865, 1996.

[131] SL Dudarev, GA Botton, SY Savrasov, CJ Humphreys, and AP Sutton. Electron- energy-loss spectra and the structural stability of nickel oxide: An lsda+ u study. Physical Review B, 57(3):1505, 1998.

99 [132] Hendrik J Monkhorst and James D Pack. Special points for brillouin-zone integrations. Physical review B, 13(12):5188, 1976.

[133] Xavier Gonze and Changyol Lee. Dynamical matrices, Born eﬀective charges, dielectric permittivity tensors, and interatomic force constants from density-functional perturbation theory. Physical Review B, 55(16):10355, 1997.

[134] Xifan Wu, David Vanderbilt, and DR Hamann. Systematic treatment of displacements, strains, and electric ﬁelds in density-functional perturbation theory. Physical Review B, 72(3):035105, 2005.

[135] Adam W Welch, Lauryn L Baranowski, Pawel Zawadzki, Stephan Lany, Colin A Wolden, and Andriy Zakutayev. Cusbse2 photovoltaic devices with 3% eﬃciency. Applied Physics Express, 8(8):082301, 2015.

[136] Adam W Welch, Lauryn L Baranowski, Haowei Peng, Hannes Hempel, Rainer Eich- berger, Thomas Unold, Stephan Lany, Colin Wolden, and Andriy Zakutayev. Trade- oﬀs in thin ﬁlm solar cells with layered chalcostibite photovoltaic absorbers. Advanced Energy Materials, 7(11), 2017.

[137] Christopher Tholander, Jens Birch, Ferenc Tasnádi, Lars Hultman, Justinas Palisaitis, PO A˚ Persson, Jens Jensen, Per Sandström, Björn Alling, and A Zukauskait˙e.ˇ Ab initio calculations and experimental study of piezoelectric yxin1−xn thin films deposited using reactive magnetron sputter epitaxy. Acta Materialia, 105:199–206, 2016.

[138] Agn˙e Zukauskait˙e,ˇ Christopher Tholander, Justinas Palisaitis, Per O A˚ Persson, Vanya Darakchieva, Nebiha Ben Sedrine, Ferenc Tasnádi, Björn Alling, Jens Birch, and Lars Hultman. Yxal1- xn thin films. Journal of Physics D: Applied Physics, 45(42):422001, 2012.

[139] JT Luo, B Fan, F Zeng, and F Pan. Inﬂuence of cr-doping on microstructure and piezoelectric response of aln ﬁlms. Journal of Physics D: Applied Physics, 42(23):235406, 2009.

[140] Ramin Matloub, Alvaro Artieda, Cosmin Sandu, Evgeny Milyutin, and Paul Muralt. Electromechanical properties of al0. 9sc0. 1n thin ﬁlms evaluated at 2.5 ghz ﬁlm bulk acoustic resonators. Applied Physics Letters, 99(9):092903, 2011.

[141] Gunilla Wingqvist, Ferenc Tasnadi, Agne Zukauskaite, Jens Birch, Hans Arwin, and Lars Hultman. Increased electromechanical coupling in w- sc x al 1- x n. Applied Physics Letters, 97(11):112902, 2010.

100 [142] M Nasr Eddine, EF Bertaut, M Roubin, and J Paris. Etude cristallographique de cr1- xvxn `abasse temperature. Acta Crystallographica Section B: Structural Crystallogra- phy and Crystal Chemistry, 33(10):3010–3013, 1977.

[143] Ralph WG Wyckoﬀ. Crystal structures (interscience, new york, 1963), vol. 1. Google Scholar, pages 796–797, 1965.

[144] VV Felmetsger and MK Mikhov. Reactive magnetron sputtering of piezoelectric cr- doped aln thin ﬁlms. In Ultrasonics Symposium (IUS), 2011 IEEE International, pages 835–839. IEEE, 2011.

[145] Linran Fan, Xiankai Sun, Chi Xiong, Carsten Schuck, and Hong X Tang. Aluminum nitride piezo-acousto-photonic crystal nanocavity with high quality factors. Applied Physics Letters, 102(15):153507, 2013.

[146] Chi Xiong, Linran Fan, Xiankai Sun, and Hong X Tang. Cavity piezooptomechan- ics: Piezoelectrically excited, optically transduced optomechanical resonators. Applied Physics Letters, 102(2):021110, 2013.

[147] Suk-Hee Park, Han Bit Lee, Si Mo Yeon, Jeanho Park, and Nak Kyu Lee. Flexible and stretchable piezoelectric sensor with thickness-tunable conﬁguration of electro- spun nanoﬁber mat and elastomeric substrates. ACS applied materials & interfaces, 8(37):24773–24781, 2016.

[148] Changhyun Pang, Gil-Yong Lee, Tae-il Kim, Sang Moon Kim, Hong Nam Kim, Sung- Hoon Ahn, and Kahp-Yang Suh. A ﬂexible and highly sensitive strain-gauge sensor using reversible interlocking of nanoﬁbres. Nature materials, 11(9):795, 2012.

[149] John A Rogers, Takao Someya, and Yonggang Huang. Materials and mechanics for stretchable electronics. science, 327(5973):1603–1607, 2010.

[150] Seon-Bae Kim, Hyejin Park, Seung-Hyun Kim, H Clyde Wikle, Jung-Hyun Park, and Dong-Joo Kim. Comparison of mems pzt cantilevers based on d 31 and d 33 modes for vibration energy harvesting. Journal of microelectromechanical{ } systems{ }, 22(1):26–33, 2013.

[151] Agn˙e Zukauskait˙e,ˇ Christopher Tholander, Justinas Palisaitis, Per O A˚ Persson, Vanya Darakchieva, Nebiha Ben Sedrine, Ferenc Tasnádi, Björn Alling, Jens Birch, and Lars Hultman. YxAl1−xN thin films. Journal of Physics D: Applied Physics, 45(42):422001, 2012.

101 [152] Yao Zhang, Weixin Zhu, Dong Zhou, Yixi Yang, and Chengtao Yang. Eﬀects of sputtering atmosphere on the properties of c-plane ScAlN thin ﬁlms prepared on sapphire substrate. Journal of Materials Science: Materials in Electronics, 26(1):472–478, 2015.

[153] Richard RA Syms, H Zou, and P Boyle. Mechanical stability of a latching MEMS variable optical attenuator. Journal of microelectromechanical systems, 14(3):529–538, 2005.

[154] Christopher Tholander, Igor A Abrikosov, Lars Hultman, and Ferenc Tasn´adi. Volume matching condition to establish the enhanced piezoelectricity in ternary (Sc, Y)0.5 (Al, Ga, In)0.5N alloys. Physical Review B, 87(9):094107, 2013.

[155] Georg Kresse and D Joubert. From ultrasoft pseudopotentials to the projector augmented-wave method. Physical Review B, 59(3):1758, 1999.

[156] Bejnamin P Burton, Axel van de Walle, and U Kattner. First principles phase diagram calculations for the wurtzite-structure systems aln–gan, gan–inn, and aln–inn. Journal of applied physics, 100(11):113528, 2006.

[157] Jonathan K Bunn, Richard Z Voepel, Zhiyong Wang, Edward P Gatzke, Jochen A Lauterbach, and Jason R Hattrick-Simpers. Development of an optimization procedure for magnetron-sputtered thin ﬁlms to facilitate combinatorial materials research. Industrial & Engineering Chemistry Research, 55(5):1236–1242, 2016.

[158] C Hums, J Bl¨asing, A Dadgar, A Diez, T Hempel, J Christen, A Krost, K Lorenz, and E Alves. Metal-organic vapor phase epitaxy and properties of AlInN in the whole compositional range. Applied Physics Letters, 90(2):022105, 2007.

[159] Aaron M Holder, Sebastian Siol, Paul F Ndione, Haowei Peng, Ann M Deml, Bethany E Matthews, Laura T Schelhas, Michael F Toney, Roy G Gordon, William Tumas, et al. Novel phase diagram behavior and materials design in heterostructural semiconductor alloys. Science advances, 3(6):e1700270, 2017.

[160] Guijuan Zhao, Xiaoqing Xu, Huijie Li, Hongyuan Wei, Dongyue Han, Zesheng Ji, Yulin Meng, Lianshan Wang, and Shaoyan Yang. The immiscibility of InAlN ternary alloy. Scientiﬁc reports, 6:26600, 2016.

[161] David Vanderbilt. Berry-phase theory of proper piezoelectric response. Journal of Physics and Chemistry of Solids, 61(2):147–151, 2000.

102 [162] Badri Narayanan, Ivar E Reimanis, Edwin R Fuller Jr, and Cristian V Ciobanu. Elastic constants of β-eucryptite studied by density functional theory. Physical Review B, 81(10):104106, 2010.

[163] Ferenc Tasnádi, Igor A Abrikosov, and Ilia Katardjiev. Significant configurational dependence of the electromechanical coupling constant of B0.125Al0.875N. Applied Physics Letters, 94(15):151911, 2009.

[164] Xinhua Yang, Guowei Zeng, and Weizhong Cao. Analysis of mechanical and electrical damages in piezoelectric ceramics. In Piezoelectric Ceramics. InTech, 2010.

[165] L Egerton and Dolores M Dillon. Piezoelectric and dielectric properties of ceramics in the system potassium-sodium niobate. Journal of the American Ceramic Society, 42(9):438–442, 1959.

[166] D Yang, LL Wei, XL Chao, ZP Yang, and XY Zhou. First-principles calculation of the eﬀects of Li-doping on the structure and piezoelectricity of (K0.5Na0.5)NbO3 lead-free ceramics. Physical Chemistry Chemical Physics, 18(11):7702–7706, 2016.

[167] Qing-Ming Wang, Xiao-Hong Du, Baomin Xu, and L Eric Cross. Electromechanical coupling and output eﬃciency of piezoelectric bending actuators. IEEE transactions on ultrasonics, ferroelectrics, and frequency control, 46(3):638–646, 1999.

[168] Moojoon Kim, Jungsoon Kim, and Wenwu Cao. Aspect ratio dependence of electromechanical coupling coeﬃcient of piezoelectric resonators. Applied Physics Letters, 87(13):132901, 2005.

[169] Gunilla Wingqvist. Thin-ﬁlm electro-acoustic sensors based on AlN and its alloys: possibilities and limitations. Microsystem technologies, 18(7-8):1213–1223, 2012.

[170] K. Uchino. Piezoelectric Actuators and Ultrasonic Motors. Electronic Materials: Sci- ence & Technology. Springer US, 1996.

[171] Shuhai Liu, Longfei Wang, Zheng Wang, Yafeng Cai, Xiaolong Feng, Yong Qin, and Zhong Lin Wang. Double-channel piezotronic transistors for highly sensitive pressure sensing. ACS Nano, 12(2):1732–1738, 2018.

[172] R. Weigel, D. P. Morgan, J. M. Owens, A. Ballato, K. M. Lakin, K. Hashimoto, and C. C. W. Ruppel. Microwave acoustic materials, devices, and applications. IEEE Transactions on Microwave Theory and Techniques, 50(3):738–749, Mar 2002.

103 [173] Wenzhuo Wu and Zhong Lin Wang. Piezotronics and piezo-phototronics for adaptive electronics and optoelectronics. Nature Reviews Materials, 1(7):16031, 2016.

[174] Yasuyoshi Saito, Hisaaki Takao, Toshihiko Tani, Tatsuhiko Nonoyama, Kazumasa Takatori, Takahiko Homma, Toshiatsu Nagaya, and Masaya Nakamura. Lead-free piezoceramics. Nature, 432(7013):84, 2004.

[175] R. Armiento, B. Kozinsky, G. Hautier, M. Fornari, and G. Ceder. High-throughput screening of perovskite alloys for piezoelectric performance and thermodynamic stability. Physical Review B, 89:134103, Apr 2014.

[176] Rickard Armiento, Boris Kozinsky, Marco Fornari, and Gerbrand Ceder. Screening for high-performance piezoelectrics using high-throughput density functional theory. Physical Review B, 84(1):014103, 2011.

[177] Maarten De Jong, Wei Chen, Henry Geerlings, Mark Asta, and Kristin Aslaug Persson. A database to enable discovery and design of piezoelectric materials. Scientiﬁc Data, 2, 2015.

[178] Liang Dong, Jun Lou, and Vivek B. Shenoy. Large in-plane and vertical piezoelectricity in janus transition metal dichalchogenides. ACS Nano, 11(8):8242–8248, 2017.

[179] Michael N Blonsky, Houlong L Zhuang, Arunima K Singh, and Richard G Hennig. Ab initio prediction of piezoelectricity in two-dimensional materials. ACS Nano, 9(10):9885–9891, 2015.

[180] Wenbin Li and Ju Li. Piezoelectricity in two-dimensional group-III monochalcogenides. Nano Research, 8(12):3796–3802, Dec 2015.

[181] Huabing Yin, Jingwei Gao, Guang Ping Zheng, Yuanxu Wang, and Yuchen Ma. Giant piezoelectric eﬀects in monolayer group-v binary compounds with honeycomb phases: A ﬁrst-principles prediction. The Journal of Physical Chemistry C, 121(45):25576– 25584, 2017.

[182] Karel-Alexander N Duerloo, Mitchell T Ong, and Evan J Reed. Intrinsic piezoelectricity in two-dimensional materials. The Journal of Physical Chemistry Letters, 3(19):2871–2876, 2012.

[183] Ronan Hinchet, Usman Khan, Christian Falconi, and Sang-Woo Kim. Piezoelectric properties in two-dimensional materials: Simulations and experiments. Materials To- day, 2018.

104 [184] R Lec and W Soluch. The elastic, piezoelectric, dielectric and acoustic properties of liio3 crystals. In Ultrasonics Symposium, 1977, pages 389–392. IEEE, 1977.

[185] Abhijit Pramanick, Dragan Damjanovic, John E. Daniels, Juan C. Nino, and Jacob L. Jones. Origins of electro-mechanical coupling in polycrystalline ferroelectrics during subcoercive electrical loading. Journal of the American Ceramic Society, 94(2):293– 309, 2011.

[186] Z. Li, M. Grimsditch, X. Xu, and S. K. Chan. The elastic, piezoelectric and dielectric constants of tetragonal pbtio3 single crystals. Ferroelectrics, 141(1):313–325, 1993.

[187] K Lefki and GJM Dormans. Measurement of piezoelectric coeﬃcients of ferroelectric thin ﬁlms. Journal of Applied Physics, 76(3):1764–1767, 1994.

[188] Andrey V Sotnikov, Hagen Schmidt, Manfred Weihnacht, Elena P Smirnova, Ta- tiana Yu Chemekova, and Yuri N Makarov. Elastic and piezoelectric properties of aln and lialo2 single crystals. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 57(4), 2010.

[189] Zuo-Guang Ye. Handbook of advanced dielectric, piezoelectric and ferroelectric materials: Synthesis, properties and applications. Elsevier, 2008.

[190] Guenter Bergerhoﬀ, R Hundt, R Sievers, and ID Brown. The inorganic crystal structure data base. Journal of Chemical Information and Computer Sciences, 23(2):66–69, 1983.

[191] Alec Belsky, Mariette Hellenbrandt, Vicky Lynn Karen, and Peter Luksch. New developments in the inorganic crystal structure database (icsd): accessibility in support of materials research and design. Acta Crystallographica Section B: Structural Science, 58(3):364–369, 2002.

[192] Prashun Gorai, Eric S Toberer, and Vladan Stevanovi´c. Computational identiﬁcation of promising thermoelectric materials among known quasi-2D binary compounds. Journal of Materials Chemistry A, 4(28):11110–11116, 2016.

[193] Michael Ashton, Joshua Paul, Susan B. Sinnott, and Richard G. Hennig. Topology- scaling identiﬁcation of layered solids and stable exfoliated 2d materials. Physical Review Letter, 118:106101, Mar 2017.

105 [194] Nicolas Mounet, Marco Gibertini, Philippe Schwaller, Davide Campi, Andrius Merkys, Antimo Marrazzo, Thibault Sohier, Ivano Eligio Castelli, Andrea Cepellotti, Giovanni Pizzi, and Nicola Marzari. Two-dimensional materials from high-throughput computational exfoliation of experimentally known compounds. Nature Nanotechnology, page 1, 2018.

[195] Gowoon Cheon, Karel-Alexander N Duerloo, Austin D Sendek, Chase Porter, Yuan Chen, and Evan J Reed. Data mining for new two-and one-dimensional weakly bonded solids and lattice-commensurate heterostructures. Nano Letters, 17(3):1915–1923, 2017.

[196] Sébastien Lebègue, T Björkman, Mattias Klintenberg, Risto M Nieminen, and Olle Eriksson. Two-dimensional materials from data filtering and ab initio calculations. Physical Review X, 3(3):031002, 2013.

[197] Georg Kresse and J¨urgen Hafner. Ab initio molecular dynamics for liquid metals. Physical Review B, 47(1):558, 1993.

[198] Georg Kresse and Jürgen Furthmüller. Efficient iterative schemes for ab initio total- energy calculations using a plane-wave basis set. Physical Review B, 54(16):11169, 1996.

[199] Ji ˇr´ı Klimeˇs, David R. Bowler, and Angelos Michaelides. Van der Waals density functionals applied to solids. Physical Review B, 83:195131, May 2011.

[200] Jiˇr´ıKlimeˇs, David R Bowler, and Angelos Michaelides. Chemical accuracy for the van der Waals density functional. Journal of Physics: Condensed Matter, 22(2):022201, 2009.

[201] Stefano Baroni, Stefano De Gironcoli, Andrea Dal Corso, and Paolo Giannozzi. Phonons and related crystal properties from density-functional perturbation theory. Reviews of Modern Physics, 73(2):515, 2001.

[202] Stefano Baroni, Paolo Giannozzi, and Andrea Testa. Greens-function approach to linear response in solids. Physical Review Letter, 58(18):1861, 1987.

[203] Xavier Gonze. Adiabatic density-functional perturbation theory. Physical Review A, 52(2):1096, 1995.

106 [204] Yvon Le Page and Paul Saxe. Symmetry-general least-squares extraction of elastic data for strained materials from ab initio calculations of stress. Physical Review B, 65(10):104104, 2002.

[205] Badri Narayanan, Ivar E Reimanis, Edwin R Fuller Jr, and Cristian V Ciobanu. Elastic constants of β-eucryptite studied by density functional theory. Physical Review B, 81(10):104106, 2010.

[206] Noa Marom, Alexandre Tkatchenko, Matthias Scheﬄer, and Leeor Kronik. Describing both dispersion interactions and electronic structure using density functional theory: The case of metal- phthalocyanine dimers. Journal of Chemical Theory and Compu- tation, 6(1):81–90, 2009.

[207] Max Born. On the stability of crystal lattices. i. In Mathematical Proceedings of the Cambridge Philosophical Society, volume 36, pages 160–172. Cambridge University Press, 1940.

[208] F´elix Mouhat and Fran¸cois-Xavier Coudert. Necessary and suﬃcient elastic stability conditions in various crystal systems. Physical Review B, 90(22):224104, 2014.

[209] Prashun Gorai, Eric S Toberer, and Vladan Stevanovi´c. Thermoelectricity in transition metal compounds: the role of spin disorder. Physical Chemistry Chemical Physics, 18(46):31777–31786, 2016.

[210] A python framework for high-throughput ﬁrst-principles calculations. https:// github.com/pylada.

[211] Jiang Yu Li. The eﬀective electroelastic moduli of textured piezoelectric polycrystalline aggregates. Journal of the Mechanics and Physics of Solids, 48(3):529–552, 2000.

[212] D. Zagorac, K. Doll, J. C. Sch¨on, and M. Jansen. Ab initio structure prediction for lead sulﬁde at standard and elevated pressures. Physical Review B, 84:045206, Jul 2011.

[213] T Chattopadhyay, J X Boucherle, and H G vonSchnering. Neutron diﬀraction study on the structural phase transition in gete. Journal of Physics C: Solid State Physics, 20(10):1431, 1987.

[214] HH Sutherland, JHC Hogg, and PD Walton. Indium polytelluride in2te5. Acta Crys- tallographica Section B: Structural Crystallography and Crystal Chemistry, 32(8):2539– 2541, 1976.

107 [215] KA Chuntonov, AN Orlov, SP Yatsenko, Yu N Grin, and LD Miroshnikova. Synthesis 1 6 of chalcogenides of a2 b3 composition and crystalline structure of rb2te3 and cs2te3. Izv. Akad. Nauk SSSR, Neorg. Mater, 18(7):1113–1116, 1982.

[216] RW Wolfe, RE Newnahm, and MI Kay. Crystal structure of bi2wo6. Solid State Communications, 7(24):1797–1801, 1969.

[217] Virgil E Bottom. Measurement of the piezoelectric coeﬃcient of quartz using the fabry-perot dilatometer. Journal of Applied physics, 41(10):3941–3944, 1970.

[218] Guillaume Boivin, Pierre B´elanger, and Ricardo J. Zednik. Torsional piezoelectric strain in monocrystalline paratellurite. In THERMEC 2016, volume 879 of Materials Science Forum, pages 637–641. Trans Tech Publications, 2 2017.

[219] Pio Baettig, Charles F Schelle, Richard LeSar, Umesh V Waghmare, and Nicola A Spaldin. Theoretical prediction of new high-performance lead-free piezoelectrics. Chemistry of Materials, 17(6):1376–1380, 2005.

[220] Pierre Hohenberg and Walter Kohn. Inhomogeneous electron gas. Physical review, 136(3B):B864, 1964.

[221] Walter Kohn and Lu Jeu Sham. Self-consistent equations including exchange and correlation eﬀects. Physical review, 140(4A):A1133, 1965.

[222] Richard M Martin. Electronic structure: basic theory and practical methods. Cambridge university press, 2004.

[223] Max Born and Robert Oppenheimer. Zur quantentheorie der molekeln. Annalen der physik, 389(20):457–484, 1927.

[224] Walter Kohn. Nobel lecture: Electronic structure of matterwave functions and density functionals. Reviews of Modern Physics, 71(5):1253, 1999.

[225] Axel D Becke. Density-functional exchange-energy approximation with correct asymp- totic behavior. Physical review A, 38(6):3098, 1988.

[226] John P Perdew and Yue Wang. Accurate and simple analytic representation of the electron-gas correlation energy. Physical Review B, 45(23):13244, 1992.

108 [227] WG Stirling, B Dorner, JDN Cheeke, and J Revelli. Acoustic phonons in the transition- metal dichalcogenide layer compound, tise2. Solid State Communications, 18(7):931– 933, 1976.

[228] JO Jenkins, JA Rayne, and RW Ure. Elastic moduli of bi2te3 from 4.2 k to 300 k. Physics Letters A, 30(6):349–350, 1969.

[229] M Gatulle, M Fischer, and A Chevy. Elastic constants of the layered compounds gas, gase, inse, and their pressure dependence i. experimental part. physica status solidi (b), 119(1):327–336, 1983.

[230] B Dorner, RE Ghosh, and G Harbeke. Phonon dispersion in the layered compound pbi2. physica status solidi (b), 73(2):655–659, 1976.

[231] Eiji Tato, Teruo Komatsu, and Yozo Kaifu. Determination of elastic constants of bii3 and sbi3 layered crystals by means of brillouin scattering. Journal of the Physical Society of Japan, 54(9):3597–3604, 1985.

[232] M Sato and K Abe. Acoustic phonon dispersion in nite2. Journal of Physics C: Solid State Physics, 12(16):L613, 1979.

[233] David Eugene Moncton, JD Axe, and FJ DiSalvo. Neutron scattering study of the charge-density wave transitions in 2 h- ta se 2 and 2 h- nb se 2. Physical Review B, 16(2):801, 1977.

[234] S Hauss¨uhl and H Scholz. Thermoelastische und elastische eigenschaften des tetrago- nalen quecksilberdijodids. Crystal Research and Technology, 10(11):1175–1179, 1975.

[235] John Sandercock. Some recent applications of brillouin scattering in solid state physics. Festk¨orperprobleme 15, pages 183–202, 1975.

[236] JL Feldman. Elastic constants of 2h-mos2 and 2h-nbse2 extracted from measured dispersion curves and linear compressibilities. Journal of Physics and Chemistry of Solids, 37(12):1141–1144, 1976.

[237] Robert E Newnham. Properties of materials: anisotropy, symmetry, structure. Oxford University Press on Demand, 2005.

109 APPENDIX A CLASSIFICATION OF PIEZOELECTRIC MATERIALS

The crystallographic symmetry of materials play crucial role on piezoelectricity. From crystallographic symmetry perspective, all the components of piezoelectric coeﬃcient tensor are zero for centrosymmetric materials. Here, we provide detail description of crystal classes from symmetry perspective and their piezoelectric behavior.

Crystal Non-centrosymmetric Centrosymmetric class Polar Non-polar Cubic m3 m3m none 432 3"m 23 4 or Te t r a gona l 4 or m 4 4mm 4" 4"2m 22 mmm

Orthorhombic mmm mm2 222

6 or 6" Hexagonal 6 or m 6 6mm 6"2m 622 mmm 3" 3"m Tr i gona l 3 3m 32

Monoclinic 2 or m 2 m none

1" Tr i c l i ni c 1 none

Figure A.1: Centrosymmetric and non-centrosymmetric (polar and non-polar) point groups. Only non-centrosymmetric point groups allow piezoelectricity.

The whole family of material systems can be classiﬁed into 32 crystallographic points groups. Out of these 32 point groups, 21 are non-centrosymmetric, i.e., crystals not having a center of symmetry. Out of these 21 point groups, except group 432, crystals containing all other point groups exhibit piezoelectric eﬀect. Out of these 20 non-centrosymmetric point

110 groups, 10 belong to polar crystals i.e., crystals which possess a unique polar axis, an axis showing different properties at the two ends. The classification of point groups based on their crystallographic symmetry is described in detail in Figure A.1. The polar crystals can be spontaneously polarized. The polarization can be compensated through external or internal conductivity or domain formation. The Spontaneous polarization is effected by the temperature. An electric charge is developed on the faces of the crystal perpendicular to the polar axis, if a change in temperature is imposed. This is known as pyroelectricity. The 10 classes of polar crystals are pyroelectric materials. In some of these pyroelectric crystals, the polarization along the polar axis can be reversed by reversing the polarity of electric field. Such crystals are called ferroelectric i.e., these are spontaneously polarized materials with reversible polarization.

111 APPENDIX B DENSITY FUNCTIONAL THEORY METHODS

This chapter brieﬂy reviews the major concepts, equations, and common approximations of density functional methods that applied throughout this thesis. The more detailed deriva- tions can be found in the original work by Kohn et al. [220, 221] and the textbook by R Martin [222].

B.1 Many-body Schr¨odinger equation

The time-independent Schr¨odinger equation can be written as follows:

HΨ= EΨ (B.1) where, Ψ(RI , ri) corresponds to the many-electron wave function with RI and ri describing the positions of the ions (I) and electrons (i) respectively. The total energy E of a system can then be evaluated by operating on Ψ with the Hamiltonian H. The system of interacting electrons (e) and nuclei (N) includes kinetic (T ) and Coulomb interaction (V ) terms is expressed by the following Hamiltonian,

H = Te + TN + VNN + Vee + VeN (B.2)

where, the electronic (Te) and nuclear (TN ) kinetic terms are expressed as follows,

ℏ2 T = 2 (B.3) e 2m ∇i e Xi ℏ2 2 TN = I (B.4) 2MI ∇ XI where, ℏ is the reduced Planck constant. The nuclear-nuclear (NN), electron-electron (ee), and the nuclear-electrons (eN) Coulom- bic interactions are expressed as,

112 2 1 ZI ZJ e VNN = (B.5) 2 rJ RI XI6=J | − | 1 e2 Vee = (B.6) 2 rj ri Xi6=j | − | 2 ZI e VeN = (B.7) − ri RI Xi,I | − |

where, me and MI describes the mass of the electron and ion respectively, ZI is the atomic

number of I-th ion, and e is the electron charge. Since, MI > >me, one can decouple the dynamics of the electrons from dynamics the nuclei. This decoupling is known as the Born-Oppenheimer approximation [223]. As a result, the quantum many body interacting Hamiltonian of condensed systems can be simpliﬁed to the so-called electronic Hamiltonian,

He = Te + VNN + Vee + VeN (B.8)

where ,the VNN is now a classical interaction, which may be approximated using a variety of approaches.

B.2 Density Functional Theory

The basic theorems of the density functional formalism were proved by Hohenberg and Kohn [220]. The key idea in density functional theory is to use the electron density n(r) as the basic variable instead of the wave function. This simpliﬁes the problem of working with 3n spatial variables for a n-electron problem to only 3.

B.2.1 Hohenberg-Kohn Theorem

In 1964, Hohenberg and Kohn published their paper on inhomogeneous electron gas [220], for density functional theory The two theorems are stated in the words of Martin [222] as follows:

Theorem 1: For any system of interacting particles in an external potential Vext(r), the potential Vext(r) is determined uniquely, except for a constant, by the ground state particle

113 density n0(r). Theorem 2: A universal functional for the energy E[n] in terms of the density n(r) can be

deﬁned, valid for any external potential Vext(r). For any particular Vext(r), the exact ground state energy of the system is the global minimum value of this functional, and the density

n(r) that minimizes the functional is the exact ground state density n0(r). The ﬁrst theorem states that all properties of a system are evaluated by the ground state

density n0(r), and the second that E[n] alone is enough to evaluate the exact ground state energy and density. The general form of the functional is

3 E[n]= T [n]+ Eint[n]+ d rVext(r)n(r)+ EII (B.9) Z where, the terms on the right side of the equation represent the kinetic energy of the electrons

(T [n]), the electron-electron interaction energy (Eint[n]), the interaction energy with an external potential in the form of Coulomb interaction with the nuclei, and the interaction

energy between the nuclei (EII ).

B.2.2 The Kohn-Sham Equations

Kohn and Sham suggested in 1965 [221] that the real system with interacting particles should be replaced with a system of non-interacting particles. This is accomplished by

replacing the purely external potential with an eﬀective potential Veff . The single particle interacting with the eﬀective potential is then described by the single particle wave function

ψj, that can be found by solving the single-particle equation [224] 1 2 + V (r) ǫ ψ (r)=0 (B.10) −2∇ eff − j j where ǫj is the eigenvalue of the non-interacting single particle, and the eﬀective potential is given by

n(r) ′ Veff (r)= Vext(r)+ ′ dr + Vxc(r) (B.11) Z r r | − |

114 where

δE [n(r)] V = xc (B.12) xc δn(r) The electron density in a system with N electrons is then calculated using the wave functions according to

N n(r)= ψ (r) 2 (B.13) | j | Xj=1 The total energy is given by the Kohn-Sham total energy functional

EKS[n]= Ts[n]+ drVext(r)n(r)+ EHartree[n]+ EII + Exc[n] (B.14) Z

Here, Vext(r) is the external potential due to the nuclei and any other external ﬁelds

(assumed to be independent of spin), EII is the interaction between the nuclei, and EHartree is the classical Coulomb interaction energy of the electron density with itself given by,

′ 1 3 3 ′ n(r)n(r ) EHartree = d rd r ′ (B.15) 2 Z Z r r | − | Although the Kohn-Sham total energy functional is exact in the form presented in eq.

(B.14), the exchange-correlation term Exc[n], where all many-body eﬀects are included, pose a problem because there is no universal form for it. There also exist approximate functionals which capture many-body eﬀects with good accuracy, e.g., the local-density approximation and the generalized gradient approximation which are described below.

B.2.3 Local Density Approximation

The local-density approximation (LDA) was ﬁrst suggested in the original paper by Kohn and Sham [221]. The basic idea comes from the observation that exchange-correlation eﬀects are to a large extent local in character. Therefore, they proposed to calculate

LDA the exchange-correlation energy Exc [n] with a simple integral over all space, where the exchange-correlation energy density is assumed to be the same as in homogeneous electron

115 hom gas with that density ǫxc [n(r)],

ELDA[n]= d3n(r)ǫhom[n(r)]. (B.16) xc Z xc To summarize the performance of LDA, it works well for covalent-, metalic-, and ionic- type bonds, even though it generally tends to over bind slightly. Although, the over binding causes gives lattice parameters which are lower than those experimentally obtained, trends in lattice parameters are well reproduced. However, for long range interactions such as Van der Waals interactions LDA does not work.

B.2.4 Generalized Gradient Approximation

The generalized gradient approximation (GGA) improves the LDA by including the absolute value of the gradient of the density with the value of n at each point. There are a number of implementations to introduce the gradient corrections [225, 226, 130] which still force the system to behave correctly in important limiting cases [222]. The work in this thesis has been performed using the implementation by Perdew, Burke, and Ernzerhof (PBE-GGA) [226, 130]. Although it tends to under bind and provide slightly larger lattice constants than experiment, PBE-GGA is better at providing reasonable geometries for the nitride systems investigated in this thesis, as compared to the LDA.

B.3 Technical implementation

The theoretical results presented in this dissertation are based on the density functional theory formalism we have outlined in this chapter as implemented the Vienna ab initio Soware Package (VASP). The technical details regarding choice of exchange-correlation functional, plane wave energy cutoﬀ, and electronic conﬁguration of valence electrons in PAWs are provided as necessary in the results chapters.

116 APPENDIX C SIMULATING ELASTIC PROPERTIES

Elastic constants provide important information on the mechanical properties of materials and on their structural stability. Under a linear elastic deformation,

σij = Cijklǫkl (C.1) Xkl

where, i, j, k, l = 1, 2, 3, σij is the stress, ǫkl is the strain, and Cijkl is the stiﬀness tensor. The stiﬀness tensor is related to the total energy of the crystal by Taylor expansion in terms of the strain components truncated to the second-order, V E(V,ǫ)= E(V )+ V σ ǫ + C ǫ ǫ + ... (C.2) 0 α α 2 αβ α β Xα Xαβ

where the Voigt’s notations are used.(α, β =1, 2,..., 6) and V0 is the equilibrium volume. The crystalline structure is assumed to be stress-free. The stiﬀness moduli are related to the strain second derivatives of the total energy:

1 ∂2E Cαβ = 0 (C.3) V ∂ǫαǫβ | The calculation of elastic constants for an arbitrary crystal then requires the ability to accurately calculate derivatives of the total energy as a function of crystal deformation. For ab-initio methods, this can be done either fully numerically, from total energy curves as a function of the applied strain for different deformations, or from strain first derivatives of the energy. In VASP, IBRION=6 and ISIF 3 allows to calculate the elastic constants. The ≤ elastic tensor is determined by performing six finite distortions of the lattice and deriving the elastic constants from the strain-stress relationship.

117 APPENDIX D MODELING OF RANDOM ALLOYS

D.1 Details for Generating SQS Structures

The disordered alloys in these systems are simulated using special quasirandom structures (SQS), which we have generated with the mcsqs code of the Alloy Theoretic Automated Toolkit (ATAT). To keep calculations eﬃcient, we generate wurtzite SQS unit cell has 32 atoms. The starting lattice constants are those of AlN; the lattice constants and angles will change after the full lattice and ionic relaxation. An example of the variation of the lattice constant with alloy concentration is shown in Figure 3.2 of the Chapter 2. The SQS match the pair correlations of the ideal disordered state up to second nearest neighbor, as well as the triplet correlations at least up to the second nearest neighbor. The key details of the procedure are presented below. Procedure: Example for 12.5%B and 12.5%Y rest Al Command:

1. corrdump l = rndstr.in ro noe nop clus 2=3.5 3=2.5 − − − − − − −

2. mcsqs rc n = 32 − −

To execute these commands, two ﬁles are necessary: rndstr.in (input for random alloys) and sqscell.out (instructions for the spatial repetition of the cell). These ﬁles have the following content: rndstr.in 3.12858812 3.12858812 5.01695523 90 90 120 1.000000 0.000000 0.000000 0.000000 1.000000 0.000000 0.000000 0.000000 1.000000 0.666667 0.333333 0.499287 Al=0.75, B=0.125, Y=0.125

118 0.333333 0.666667 0.999287 Al=0.75, B=0.125, Y=0.125 0.666667 0.333333 0.880713 N 0.333333 0.666667 0.380713 N

sqscell.out 1 200 020 002

D.2 Dispersion of e33 and C33 values for diﬀerent Y concentration

400 4.0 individual-SQS individual-SQS 350 average-SQS 3.5 average-SQS

300 ) 3.0 2

(GPa) 250 2.5 (C/m 33 33 C e

200 2.0

150 1.5

100 1.0 0 10 20 30 40 50 0 10 20 30 40 50 Y (%) Y (%)

Figure D.1: Calculated e33 and C33, for each individual SQS and averaged, at each concentration.

At every alloy concentration, all properties (i.e. lattice constants, piezoelectric tensor components, elastic tensor component) were calculated for five different SQS structures, and for each property the average between the five SQS calculations was reported. The dispersion of values is a function of the composition. The reason is that there are more possible combinations for, e.g., AlN-YN structures as the YN concentration is increased. As

119 and example of dispersion, Figure D.1 shows the individual and average values for e33 and

C33 at selected Y concentrations up to 50%.

120 APPENDIX E ADDITIONAL SUPPLEMENTARY INFORMATION OF CHAPTER 3

E.1 Variations of e31 and C11 with Y and B composition

In the main text, we discussed the e33 and C33 coefficients which are important for thin film bulk acoustic wave resonators (TFBARs) in sensor, transducer, and actuator applications. Piezoelectric materials in the bending mode, e.g., contracting actuators, tube actuators, or bending actuators, the important piezoelectric and stiffness coefficients are e31 and

C11. For completeness, these are plotted in Figure E.1 as a function of the Y concentration, at ﬁxed boron levels.

E.2 Piezoelectric moduli as functions of composition

The piezoelectric moduli are calculated from,

e33(C11 + C12) 2e31C12 d33 = − 2 (E.1) (C11 + C12)C33 2C − 13 e31C33 e33C13 d31 = − 2 (E.2) (C11 + C12)C33 2C − 13 and are plotted in Figure E.2.

121 Figure E.1: Piezoelectric coeﬃcient e31 and elastic modulus C11 as functions of Y and B concentrations.

122 Figure E.2: Calculated d33 and d31 surface in yttrium-boron composition space

123 APPENDIX F SUPPLEMENTARY INFORMATION OF CHAPTER 4

This chapter contains the supplemental information of Chapter 4.

F.1 Comparison of Experimental and Calculated Piezoelectric Modulus

Using density functional theory, we have calculated longitudinal piezoelectric modulus

(d33) of well-known piezoelectric compounds. The predicted modulus are found to be in good agreement with experimentally measured values. Since, they are all exist in bulk 3D crystal, standard GGA-PBE exchange correlation functionals have been used for these calculations. Plane wave cutoﬀ energy of 540 eV is used for structural relaxation, calculation of elastic

tensors and piezoelectric coeﬃcient tensor. k-point grid, deﬁned by natoms nkpoints 1000, × ≈ where natoms is number of atoms in the primitive cell and nkpoints is the number of k-points, is used throughout the calculations. A very high tolerance of 10−8 eV for energy convergence is used for all three calculations.

BaTiO3

100 PbTiO3 (pC/N) 33 d

ZnO 20% error 10 CdS CdSe

AlN GaN GGA-Calculated GGA-Calculated

1 1 10 100

Experimental d33 (pC/N)

Figure F.1: Calculated d33 with GGA functional vs experimentally measured d33

124 F.2 Axial Anisotropies: C vs e

We have investigated all the possible correlations among the axial anisotropies of d, e, and C. In the main text, we have described the axial anisotropy dependence of d on the axial anisotropies of C and e. The remaining plot between axial anisotropy of C and e is shown in Figure F.2.

NaN3 10 CuVO3 ) 22 Li SbO +C 7 6 Bi2WO6 In2Te5 11 1 /(C MoSe2 33 GeTe C WS2 0.1 BN

1 10

e33/(e11+e22)

Figure F.2: Axial anisotropy in C vs axial anisotropy in e

F.3 Experimentally Measured Elastic Constants of Quasi-2D Materials

Table F.1: Reference table for Figure 6. The measured values of Cij are reported in GPa unit

Material SG C11 C12 C33 C44 Technique Temperature

TiSe2 [227] 164 120 42 39 14.3 NeutronInelasticScattering 300K Bi2Te3 [228] 166 74 - 52 31.35 WaveResonanceTechnique 4.2K GaSe[229] 187 103 28.9 34.1 9 BrillouinScattering 300K InSe[229] 194 73 27 36 11.7 BrillouinScattering 300K GaS[229] 194 123.3 34.7 38.2 9.96 BrillouinScattering 300K PbI2 [230] 166 27.7 9.6 20.2 6.2 NeutronInelasticScattering 293K BiI3 [231] 162 29 5 26 7 BrillouinScattering 83K NiTe2 [232] 164 109.5 41.9 52.6 20.4 NeutronSpectrometry 300K TaSe2 [233] 194 229 36.6 54 18.5 NeutronSpectrometry 300K NbSe2 [233] 194 194 91 42 17.6 NeutronSpectrometry 300K HgI2 [234] 137 33 5.6 16.3 7.23 UltrasoundPropagationTechnique 293K CdI2 [235] 164 41.21 16 21.8 18.6 BrillouinScattering - SnS2 [235] 164 124 33 28.9 10.2 BrillouinScattering - MoS2 [236] 194 238 -54 51 21.7 NeutronInelasticScattering 308K

125 F.4 Schematics of Diﬀerent Modes of d

휎 휎 휎

V V V d d 11-mode, d11 12-mode, 12 13-mode, 13

휎 휎 휎 V V V d 14-mode, 14 d 15-mode, d15 16-mode, 16 휎 휎 휎

V V V d d 21-mode, d21 22-mode, 22 23-mode, 23

휎 휎 휎

V V V d 24-mode, 24 d 25-mode, d25 26-mode, 26 휎 휎

휎 V V V d d 31-mode, d31 32-mode, 32 33-mode, 33

휎 휎 휎 V V V d 34-mode, 34 d 35-mode, d35 36-mode, 36

Figure F.3: Schematics of the modes of d with their measuring polarization directions and deformation stresses. Each mode represents each component of d-matrix

126 F.5 Rotation of d, e, and C -matrices

In our reported 63 quasi-2D piezoelectric materials, we found that in 19 candidates the layer stacking direction is not along [001]-direction. To maintain consistency and capture the eﬀect of van der Waals bond on piezoelectric response, we rotate their d, e, and C-matrix into the co-ordinate frame such that the layering direction is always along [001]-direction. The matrix transformation in going from the old to the new coordinate system[15, 237] is,

0 x3 x3 (a) (b) 0old0 x x x 0 1 2 3 x1 a11 a12 a13

0 0 x2 a21 a22 a23

−1 new cos a23 0 0 0 x x3 a31 a32 a33 −1 2 cos a22 −1 cos a21 x1 x2 0 x1

Figure F.4: (a) ‘old’ and ‘new’ co-ordinate frames. (b) The angular relations between the axes is speciﬁed by drawing up a table of direction cosines which is denoted by a-matrix

′ e =(a)(e)(α−1) (F.1) ′ d =(a)(d)(α−1) (F.2) ′ C =(α)(C)(αt) (F.3)

Where, a is the direction cosine matrix, given by,

a11 a12 a13 a21 a22 a23 a31 a32 a33   Elements of α is given by,

2 2 2 (a11) (a12) (a13) (2a12a13) (2a13a11) (2a11a12) 2 2 2  (a21) (a22) (a23) (2a22a23) (2a23a21) (2a21a22)  (a2 ) (a2 ) (a2 ) (2a a ) (2a a ) (2a a )  31 32 33 32 33 33 31 31 32  (a21a31) (a22a32) (a23a33) (a22a33)+(a23a32) (a21a33 + a23a31) (a22a31 + a21a32)   (a31a11) (a32a12) (a33a13) (a12a33)+(a13a32) (a13a31 + a11a33) (a11a32 + a12a31)   (a11a21) (a12a22) (a13a23) (a12a23)+(a13a22) (a13a21 + a11a23) (a11a22 + a12a21)  

127 For detail derivation of these transformations please see reference [15, 237].

128 APPENDIX G PERMISSIONS FOR CONTENT REUSED

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Geoff Brennecka Mon, Jul 23, 2018 at 12:51 PM To: Sukriti Manna

I, Geoff Brennecka, hereby authorize Sukriti Manna to use any part or all of the article entitled "Tuning the piezoelectric and mechanical properties of the AlN system via alloying with YN and BN" published in Journal of Applied Physics (https://doi.org/10.1063/1.4993254) in his published dissertation. [Quoted text hidden]

Vladan Stevanovic Mon, Jul 23, 2018 at 12:52 PM To: Sukriti Manna , Cristian Ciobanu , Geoff Brennecka

I, Vladan Stevanovic, hereby authorize Sukriti Manna to use any part or all of the article entitled "Tuning the piezoelectric and mechanical properties of the AlN system via alloying with YN and BN" published in Journal of Applied Physics (https://doi.org/10.1063/1.4993254) in his published dissertation.

Best,

Vladan

From: Sukriti Manna Date: Monday, July 23, 2018 at 12:23 PM To: Cristian Ciobanu , Vladan Stevanovic , Geoff https://mail.google.com/mail/u/1/?ui=2&ik=66636edd75&jsver=EWKsbuuUcyk.en.&cbl=gmail_fe_180716.14_p7&view=pt&search=inbox&th=164ca959abd2992c… 1/2

133 134 7/23/2018 Colorado School of Mines Mail - Permission to Reuse Submitted JMCC paper for Dissertation

Sukriti Manna

Permission to Reuse Submitted JMCC paper for Dissertation 5 messages

Sukriti Manna Mon, Jul 23, 2018 at 12:25 PM To: Prashun Gorai , Geoff Brennecka , cciobanu , Vladan Stevanovic

Dear All,

I am uploading my Ph.D. thesis, and the graduate school requires the permissions from coauthors on a published paper that used in the thesis. Could you please reply to this email with your permissions on our paper "Large Piezoelectric Response of van der Waals Layered Solids" submitted to Journal of Materials Chemistry C?

Please use this specific wording for the permission: I, YOUR NAME, hereby authorize Sukriti Manna to use any part or all of the manuscript entitled "Large Piezoelectric Response of van der Waals Layered Solids" submitted to Journal of Materials Chemistry C (ID: TCART052018002560) in his published dissertation.

It will be great if I can get your permissions by today or as soon as possible. Thank you so much for helping.

Sukriti

Geoff Brennecka Mon, Jul 23, 2018 at 12:51 PM To: Sukriti Manna

I, Geoff Brennecka, hereby authorize Sukriti Manna to use any part or all of the manuscript entitled "Large Piezoelectric Response of van der Waals Layered Solids" submitted to Journal of Materials Chemistry C (ID: TCART052018002560) in his published dissertation. [Quoted text hidden]

Vladan Stevanovic Mon, Jul 23, 2018 at 12:51 PM To: Sukriti Manna , Prashun Gorai , Geoff Brennecka , Cristian Ciobanu

Hi Sukriti,

I, Vladan Stevanovic, hereby authorize Sukriti Manna to use any part or all of the manuscript entitled "Large Piezoelectric Response of van der Waals Layered Solids" submitted to Journal of Materials Chemistry C (ID: TCART052018002560) in his published dissertation.

Best,

Vladan

From: Sukriti Manna Date: Monday, July 23, 2018 at 12:26 PM https://mail.google.com/mail/u/1/?ui=2&ik=66636edd75&jsver=EWKsbuuUcyk.en.&cbl=gmail_fe_180716.14_p7&view=pt&search=inbox&th=164ca968a472c8ad… 1/2

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